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The analysis and design of attitude control systems for spacecraft employing pulse-operated (on-off) thrusters is usually accomplished through a combination of modeling approximations and empirical techniques. In this paper a new thruster pulse-modulation scheme for pointing and tracking applications is developed from nonlinear control theory.
A controlled, laser-generated, freestream perturbation was created in the freestream of the Boeing/AFOSR Mach-6 Quiet Tunnel (BAM6QT). The freestream perturbation convected downstream in the Mach-6 wind tunnel to interact with a flared cone model. The geometry of the flared cone is a body of revolution bounded by a circular arc with a 3-meter radius. Fourteen PCB 132A31 pressure transducers were used to measure a wave packet generated in the cone boundary layer by the freestream perturbation. This wave packet grew large and became nonlinear before experiencing natural transition in quiet flow. Breakdown of this wave packet occurred when the amplitude of the pressure fluctuations was approximately 10% of the surface pressure for a nominally sharp nosetip. The initial amplitude of the second mode instability on the blunt flared cone is estimated to be on the order of 10 −6 times the freestream static pressure. The freestream laser-generated perturbation was positioned upstream of the model in three different configurations: on the centerline, offset from the centerline by 1.5 mm, and offset from the centerline by 3.0 mm. When the perturbation was offset from the centerline of a blunt flared cone, a larger wave packet was generated on the side toward which the perturbation was offset. The offset perturbation did not show as much of an effect on the wave packet on a sharp flared cone as it did on a blunt flared cone.
The salience of nonlinearities in the Arctic stratospheric response to El Niño - Southern Oscillation (ENSO) is evaluated in a 41 member ensemble of the period 1980 to 2009 in the NASA Goddard Earth Observing System Chemistry-Climate Model. Central Pacific El Niño (CP EN) events preferentially increase zonal wavenumber 2, while Eastern Pacific El Niño (EP EN) events preferentially increase zonal wavenumber 1, though these nonlinearities become statistically robust only after more than 45 events are considered. This leads to a difference in the morphology of SSW events preferentially forced by each ENSO flavor. While the zonal-mean Arctic stratospheric response is little different between EP and CP EN once one accounts for the relative weakness of CP events, the tropospheric subpolar negative North Atlantic Oscillation response is stronger for CP events. Other nonlinearities are even less salient: there is no indication of any nonlinearities in the wave driving between moderate EP EN and extreme EP EN, nor between EN and La Niña (LN). However in approximately 15% of the integrations within our ensemble, there is no difference in SSW frequency between EN and LN, suggesting that a similar SSW frequency for both EN and LN, as has occurred over the past 60 years, can occur by chance. The intra-ensemble variability in LN SSW frequency can be related back to intra-ensemble variability in the tropospheric response to LN.
Because of planetary encounters, the motion of near-Earth asteroids is chaotic and small differences in the initial conditions tend to diverge exponentially. Linear approximations for propagating orbital uncertainties can lead to inaccurate estimates of the probability of an Earth collision. We present a novel fully nonlinear strategy for estimating the probability of an asteroid impact using sequential Monte Carlo layers. The method first explores a low-resolution layer to locate potentially relevant regions. Then, we conduct localized searches on deeper layers with higher resolution. The method retains the accuracy of brute-force Monte Carlo sampling while reducing the computational cost by only sampling relevant regions.
The efficiency gains obtained using higher-order implicit Runge-Kutta schemes as compared with the second-order accurate backward difference schemes for the unsteady Navier-Stokes equations are investigated. Three different algorithms for solving the nonlinear system of equations arising at each timestep are presented. The first algorithm (NMG) is a pseudo-time-stepping scheme which employs a non-linear full approximation storage (FAS) agglomeration multigrid method to accelerate convergence. The other two algorithms are based on Inexact Newton's methods. The linear system arising at each Newton step is solved using iterative/Krylov techniques and left preconditioning is used to accelerate convergence of the linear solvers. One of the methods (LMG) uses Richardson's iterative scheme for solving the linear system at each Newton step while the other (PGMRES) uses the Generalized Minimal Residual method. Results demonstrating the relative superiority of these Newton's methods based schemes are presented. Efficiency gains as high as 10 are obtained by combining the higher-order time integration schemes with the more efficient nonlinear solvers.
ATIC (Advanced Thin Ionization Calorimeter) is a balloon borne experiment designed to measure the cosmic ray composition for elements from hydrogen to iron and their energy spectra from approx.50 GeV to near 100 TeV. It consists of a Si-matrix detector to determine the charge of a CRT particle, a scintillator hodoscope for tracking, carbon interaction targets and a fully active BGO calorimeter. ATIC had its first flight from McMurdo, Antarctica from 28/12/2000 to 13/01/2001. The ATIC flight collected approximately 25 million events. The silicon matrix of the ATIC spectrometer is designed to resolve individual elements from proton to iron. To provide this resolution careful calibration of each pixel of the silicon matrix is required. Firstly, for each electronic channel of the matrix the pedestal value was subtracted taking into account its drift during the flight. The muon calibration made before the flight was used then to convert electric signals (in ADC channel number) to energy deposits in each pixel. However, the preflight muon calibration was not accurate enough for the purpose, because of lack of statistics in each pixel. To improve charge resolution the correction was done for the position of Helium peak in each pixel during the flight . The other way to set electric signals in electronics channels of the Si-matrix to one scale was correction for electric channel gains accurately measured in laboratory. In these measurements it was found that small different nonlinearities for different channels are present in the region of charge Z > 20. The correction for these non-linearities was not done yet. In linear approximation the method provides practically the same resolution as muon calibration plus He-peak correction. For searching a pixel with the signal of primary particle an indication from the cascade in the calorimeter was used. For this purpose a trajectory was reconstructed using weight centers of energy deposits in BGO layers. The point of intersection of this trajectory with Si-matrix and its RMS was determined. The pixel with maximal signal in 3sigma region was taken as sought. The signal in this pixel was corrected by trajectory zenith angle. The preliminary results on charge resolution of the Si-matrix in the range from protons to iron are presented.
The structure of certain axially symmetric circulations in a stably stratified, differentially heated, rotating Boussinesq fluid on a sphere is analyzed. A simple approximate theory (similar to that introduced by Schneider (1977)) is developed for the case in which the fluid is sufficiently inviscid that the poleward flow in the Hadley cell is nearly angular momentum conserving. The theory predicts the width of the Hadley cell, the total poleward heat flux, the latitude of the upper level jet in the zonal wind, and the distribution of surface easterlies and westerlies. Fundamental differences between such nearly inviscid circulations and the more commonly studied viscous axisymmetric flows are emphasized. The theory is checked against numerical solutions to the model equations.
Here, we formulate a nonlinear optimal control problem for intraday operation of a natural gas pipeline network that includes storage reservoirs. The dynamics of compressible gas flow through pipes, compressors, reservoirs, and wells are considered. In particular, a reservoir is modeled as a rigid, hollow container that stores gas under isothermal conditions and uniform density, and a well is modeled as a vertical pipe. For each pipe, flow dynamics are described by a coupled partial differential equation (PDE) system in density and mass flux variables, with momentum dissipation modeled using the Darcy–Wiesbach friction approximation. Compressors are modeled as scaling up the pressure of gas between the inlet and outlet. The governing equations for all network components are spatially discretized and assembled into a nonlinear differential-algebraic equation (DAE) system, which synthesizes above-ground pipeline and subsurface reservoir dynamics into a single reduced-order model. We seek to maximize an objective function that quantifies economic profit and network efficiency subject to the flow equations and inequalities that represent operating limitations. The problem is solved using a primal–dual interior point solver, and the solutions are validated in computational experiments and simulations on several pipeline test networks to demonstrate the effectiveness of the proposed methodology.
The multi-harmonic balance method combined with numerical continuation provides an efficient framework to compute a family of time-periodic solutions, or response curves, for large-scale, nonlinear mechanical systems. The predictor and corrector steps repeatedly solve a sequence of linear systems that scale by the model size and number of harmonics in the assumed Fourier series approximation. In this paper, a novel Newton–Krylov iterative method is embedded within the multi-harmonic balance and continuation algorithm to efficiently compute the approximate solutions from the sequence of linear systems that arise during the prediction and correction steps. Further, the method recycles, or reuses, both the preconditioner and the Krylov subspace generated by previous linear systems in the solution sequence. A delayed frequency preconditioner refactorizes the preconditioner only when the performance of the iterative solver deteriorates. The GCRO-DR iterative solver recycles a subset of harmonic Ritz vectors to initialize the solution subspace for the next linear system in the sequence. The performance of the iterative solver is demonstrated on two exemplars with contact-type nonlinearities and benchmarked against a direct solver with traditional Newton–Raphson iterations.
The general rotorcraft aeromechanical stability program (GRASP) was developed to calculate aeroelastic stability for rotorcraft in hovering flight, vertical flight, and ground contact conditions. GRASP is described in terms of its capabilities and its philosophy of modeling. The equations of motion that govern the physical system are described, as well as the analytical approximations used to derive them. The equations include the kinematical equation, the element equations, and the constraint equations. In addition, the solution procedures used by GRASP are described. GRASP is capable of treating the nonlinear static and linearized dynamic behavior of structures represented by arbitrary collections of rigid-body and beam elements. These elements may be connected in an arbitrary fashion, and are permitted to have large relative motions. The main limitation of this analysis is that periodic coefficient effects are not treated, restricting rotorcraft flight conditions to hover, axial flight, and ground contact. Instead of following the methods employed in other rotorcraft programs. GRASP is designed to be a hybrid of the finite-element method and the multibody methods used in spacecraft analysis. GRASP differs from traditional finite-element programs by allowing multiple levels of substructure in which the substructures can move and/or rotate relative to others with no small-angle approximations. This capability facilitates the modeling of rotorcraft structures, including the rotating/nonrotating interface and the details of the blade/root kinematics for various types. GRASP differs from traditional multibody programs by considering aeroelastic effects, including inflow dynamics (simple unsteady aerodynamics) and nonlinear aerodynamic coefficients.
Hydration forces acting between macroscopic bodies at distances L ≤ 3 nm in pure water are calculated based on the phenomenological model of polar liquids. It is shown that depending on the properties of the bodies, the interacting surfaces polarize the liquid differently, and wetting properties of the surfaces are completely characterized by two parameters. If the surfaces are hydrophilic, liquid molecules are polarized at right angles to the surfaces, and the interaction is the short-range repulsion (the forces of interaction decrease exponentially over the characteristic length λ ≈ 0.2 nm). The interaction between the hydrophobic surfaces is more diversified and has been studied less. For L ≤ 3 nm, the interaction exhibits universal properties, while for L ≤ 3 nm, it considerably depends on the properties of the surfaces and on the distances between them, as well as on the composition of the polar liquid. In full agreement with the available experimental results we find that if the interfaces are mostly hydrophobic, then the interaction is attractive and long-range (the interaction forces diminish exponentially with decay length 1.2 nm). In this case, the resultant polarization of water molecules is parallel to the surface. It is shown that hydration forces are determined by nonlinear effects of polarization of the liquid in the bulk or by analogous nonlinearity of the interaction of water with a submerged body. This means that the forces of interaction cannot be calculated correctly in the linear response approximation. The forces acting between hydrophobic or hydrophilic surfaces are of the entropy type or electrostatic, respectively. It is shown that hydrophobic and hydrophilic surfaces for L ≤ 3 nm repel each other. The calculated intensity of their interaction is in agreement with experimental data. We predict the existence of an intermediate regime in which a body cannot order liquid molecules, which results in a much weaker attraction that decreases in inverse proportion to the squared distance between the surfaces of bodies. The difference between the microscopic structures of liquids confined in nanovolumes from liquids in large volumes is considered. The proposed model is applicable for a quantitative description of the properties of water at temperatures T satisfying the condition 0 < (T–T{sub c})/T{sub c} ≪ 1, where T{sub c} ≈ 230 K is the temperature of the ferroelectric phase transition observed in supercooled water. Under standard conditions, the model can be used for obtaining qualitative estimates.
Multigrid is a highly scalable class of methods most often used for solving large linear systems. In this paper we develop a nonlinear algebraic multigrid framework for the power flow equations, a complex quadratic system of the form ${diag}({v})\overline{Y{v}}={s}$, where $Y$ is approximately a complex scalar rotation of a real graph Laplacian. This is a standard problem that needs to be solved repeatedly during power grid simulations. A key difference between our multigrid framework and typical multigrid approaches is the use of a novel multiplicative coarse-grid correction to enable a dynamic multigrid hierarchy. We also develop a new type of smoother that allows one to coarsen together the different types of nodes that appear in power grid simulations. In developing a specific multigrid method, one must make a number of choices that can significantly affect the method's performance, such as how to construct the restriction and interpolation operators, what smoother to use, and how aggressively to coarsen. In this paper, we make simple but reasonable choices that result in a scalable and robust power flow solver. Experiments demonstrate this scalability and show that it is significantly more robust to poor initial guesses than current state-of-the-art solvers.
We present a novel numerical method for solving the anisotropic diffusion equation in magnetic fields confined to a periodic box which is accurate and provably stable. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing both boundary conditions and parallel diffusion operator obtained by field line tracing. We prove that the fully-discrete approximation is unconditionally stable. Discrete energy estimates are shown to match the continuous energy estimate given the correct choice of penalty parameters. A nonlinear penalty parameter is shown to provide an effective method for tuning the parallel diffusion penalty and significantly minimises rounding errors. Several numerical experiments, using manufactured solutions, the “NIMROD benchmark” problem and a single island problem, are presented to verify numerical accuracy, convergence, and asymptotic preserving properties of the method. Finally, we present a magnetic field with chaotic regions and islands and show the contours of the anisotropic diffusion equation reproduce key features in the field.
Abstract Optical phase conjugation (OPC) is a nonlinear technique used for counteracting wavefront distortions, with applications ranging from imaging to beam focusing. Here, we present a diffractive wavefront processor to approximate all-optical phase conjugation. Leveraging deep learning, a set of diffractive layers was optimized to all-optically process an arbitrary phase-aberrated input field, producing an output field with a phase distribution that is the conjugate of the input wave. We experimentally validated this wavefront processor by 3D-fabricating diffractive layers and performing OPC on phase distortions never seen during training. Employing terahertz radiation, our diffractive processor successfully performed OPC through a shallow volume that axially spans tens of wavelengths. We also created a diffractive phase-conjugate mirror by combining deep learning-optimized diffractive layers with a standard mirror. Given its compact, passive and multi-wavelength nature, this diffractive wavefront processor can be used for various applications, e.g., turbidity suppression and aberration correction across different spectral bands.
The paradigm of the inflationary universe provides a possible explanation for several observed cosmological properties. In order for such solutions to be successful, the universe must convert the energy stored in the inflaton potential into standard model particles through a process known as reheating. In this paper, we reconsider the reheating process for the case where the inflaton potential respects an approximate (but spontaneously broken) conformal symmetry during the reheating epoch. After reviewing the Effective Field Theory of Reheating, we present solutions for the nonlinear oscillations of the inflaton field, derive the corresponding Hill’s equation for the coupled reheating field, and determine the stability diagram for parametric resonance. For this class of models —the simplest realization being a scalar field with a quartic term—the expansion of the universe drives the coupled field toward a more unstable part of parameter space, in contrast to the standard case. We also generalize this class of models to include quadratic breaking terms in the potential during the reheating epoch and address the process of stability in that universality class of models.
Dynamic simulation is vitally important in power system analysis, but traditional approaches based on numerical integration over small time steps are time-consuming. Also, the Newton-Raphson method suffers from difficulty in convergence when solving nonlinear algebraic equations. In this paper, we propose a novel dynamic simulation approach based on holomorphic embedding. By obtaining a high-order approximation of system dynamics, it achieves a much larger time step and thus enhances the computational efficiency significantly. In addition, the new approach avoids non-convergence issues in solving algebraic equations, which improves robustness. The approach includes flexible modeling of synchronous generators and controllers, and we propose a method for modeling generator coordinate transformations. The approach is tested on the IEEE 39-bus, 10-generator system and a Polish 2383-bus, 327-generator system. The results demonstrate promising computational efficiency and satisfactory numerical robustness for the analysis of large-scale power systems.
Research was undertaken to determine the mass required to protect a surface during atmospheric reentry by injecting a cold gas tangentially along the surface. A consideration of this problem revealed that the boundary layer equations could be employed to approximate the most important flow configurations which occur in film cooling. However, even with this simplification, the equations to be solved are complex, nonlinear, coupled partial differential equations. Due to the relative weakness of classical methods of boundary layer analysis, finite difference methods were chosen as the best approach to the formidable problem under consideration. Instead of blindly adopting a proposed technique and trying to force it to the problem, a more systematic study was deemed to be not only desirable but necessary. A relatively basic study of the application of finite difference methods was undertaken with the objective of developing an algorithm capable of obtaining accurate solutions to film cooling problems.
A technique is described for the efficient numerical solution of nonlinear partial differential equations by rapid iteration. In particular, a special approach is described for applying the Aitken acceleration formula (a simple Pade approximant) for accelerating the iterative convergence. The method finds the most appropriate successive approximations, which are in a most nearly geometric sequence, for use in the Aitken formula. Simple examples are given to illustrate the use of the method. The method is then applied to the mixed elliptic-hyperbolic problem of steady, inviscid, transonic flow over an airfoil in a subsonic free stream.