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NASA NTRS1993

Control of nonlinear systems represented in quasilinear form

Methods to synthesize controllers for nonlinear systems are developed by exploiting the fact that under mild differentiability conditions, systems of the form: x-dot = f(x) + G(x)u can be represented in quasilinear form, viz: x-dot = A(x)x + B(x)u. Two classes of control methods are investigated. The first is zero-look-ahead control, where the control input depends only on the current values of A(x) and B(x). For this case the control input is computed by continuously solving a matrix Riccati equation as the system progresses along a trajectory. The second is controllers with look-ahead, where the control input depends on the future behavior of A(x) and B(x). These controllers use the similarity between quasilinear systems and linear time varying systems to find approximate solutions to optimal control type problems. The methods that are developed are not guaranteed to be globally stable. However in simulation studies they were found to be useful alternatives for synthesizing control laws for a general class of nonlinear systems.

Coetsee, Josef A.↗

DOE OSTI2019

A moment approach for entropy solutions to nonlinear hyperbolic PDEs

We propose to solve hyperbolic partial differential equations(PDEs) with polynomial flux using a convex optimization strategy. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is then illustrated on the celebrated Burgers equation. Here, we also compare our results with an existing numerical scheme, namely the Godunov scheme.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2019

A multi-dimensional, moment-accelerated deterministic particle method for time-dependent, multi-frequency thermal radiative transfer problems

Thermal Radiative Transfer (TRT) is the dominant energy transfer mechanism in high-energy density physics with applications in inertial confinement fusion and astrophysics. The stiff interactions between the material and radiation fields make TRT problems challenging to model. In this study, we propose a multi-dimensional extension of the deterministic particle (DP) method. The DP method combines aspects from both particle and deterministic methods. If the emission source is known a priori, and no physical scattering is present, the intensity of a particle can be integrated analytically. This introduces no statistical noise compared to Monte-Carlo methods, while maintaining the flexibility of particle methods. The method is closely related to the popular method of long characteristics. The combination of the DP-method with a discretely-consistent, nonlinear, gray low-order system enables an efficient solution algorithm for multi-frequency TRT problems. Here, we demonstrate with numerical examples that the use of a linear-source approximation based on spatial moments improves the behavior of our method in the thick diffusion limit significantly.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2025

Constraints on primordial non-Gaussianity from the cross-correlation of DESI luminous red galaxies and Planck CMB lensing

We use the angular cross-correlation between a luminous red galaxy (LRG) sample from the Dark Energy Spectroscopic Instrument (DESI) Legacy Survey data release DR9 and the Planck cosmic microwave background (CMB) lensing maps to constrain the local primordial non-Gaussianity parameter, f NL , using the scale-dependent galaxy bias effect. The galaxy sample covers approximately 40% of the sky, contains galaxies up to redshift z ∼ 1.4, and is calibrated with the LRG spectra that have been observed for DESI Year 1 (Y1). We apply a nonlinear imaging systematics treatment based on neural networks to remove observational effects that could potentially bias the f NL measurement. Our measurement is performed without blinding, but the full analysis pipeline is tested with simulations including systematics. Using the two-point angular cross-correlation between LRG and CMB lensing only, we find f NL = 39 −38 +40 at the 68% confidence level, and our result is robust in terms of systematics and cosmological assumptions. If we combine this information with the autocorrelation of LRG, applying a scale cut to limit the impact of systematics, we find f NL = 24 −21 +20 at the 68% confidence level. Our results motivate the use of CMB lensing cross-correlations to measure f NL with future datasets, given its stability in terms of observational systematics compared to the angular autocorrelation. Furthermore, performing accurate systematics mitigation is crucially important in order to achieve competitive constraints on f NL from CMB lensing cross-correlation in combination with the tracers’ autocorrelation.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2021

Structural color from a coupled nanowire pair beyond the bonding and antibonding model

Optical resonances in nanostructures can be harnessed to produce a wide range of structural colors. Conversely, the analysis of structural colors has been used to clarify the nature of optical resonances. Here, we show that silicon nanowire (NW) pairs can display a wide range of structural colors by controlling their radiative coupling. This is accomplished by exciting a series of Fabry–Pérot-like modes where light is repeatedly scattered between two NWs. These modes are beyond the expectation from the conventional chemical bonding model under a quasi-electrostatic approximation, in which only bonding and antibonding modes can be formed in a pair system through modal hybridization. The additional eigenmodes found in a two-resonator system originate from the nonlinear, frequency-dependent coupling strength derived from the radiative nature of low- Q resonators. The Fabry–Pérot modes can be tuned across the entire visible frequency range by varying the distance between two NWs, leading to what we believe is a new type of universal building blocks that can provide structural color within a subwavelength footprint. The presented results pave the way toward the design and usage of highly tunable resonances that exploit the radiative coupling of high-index nanostructures.

Li, Qitong↗

DOE OSTI2021

Large-Scale Welding Process Simulation by GPU Parallelized Computing

The computational design of industrially relevant welded structures is extremely time consuming due to coupled physics and high nonlinearity. Previously, most welding distortion and residual stress simulations have been limited to small coupons and reduced order (from three-dimensional [3D] to two-dimensional [2D]), or inherent strain approximations were used for large structures. In this current study, an explicit finite element code based on a graphics processing unit was utilized to perform 3D transient thermomechanical simulation of structural components during welding. Laser brazing of aluminum alloy panels as representative of automotive manufacturing scenarios was simulated to predict out-of-plane distortion under different clamping conditions. The predicted deformation pattern and magnitude were validated by laser scanning data of physical assemblies. In addition, the code was used to investigate residual stresses developed during multipass arc welding of a nuclear industry pressurizer surge nozzle and subsequent welding repair where a 3D simulation was necessary. Taking the experimental data as reference, the 3D model predicted better residual stress distribution than a typical 2D asymmetrical model. Stress evolution in welding repair was also presented and discussed in this study. Furthermore, the efficient numerical model made it feasible to use integrated computational welding engineering to simulate welding processes for large-scale structures.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1974

Quantum mechanical streamlines. I - Square potential barrier

Exact numerical calculations are made for scattering of quantum mechanical particles hitting a square two-dimensional potential barrier (an exact analog of the Goos-Haenchen optical experiments). Quantum mechanical streamlines are plotted and found to be smooth and continuous, to have continuous first derivatives even through the classical forbidden region, and to form quantized vortices around each of the nodal points. A comparison is made between the present numerical calculations and the stationary wave approximation, and good agreement is found between both the Goos-Haenchen shifts and the reflection coefficients. The time-independent Schroedinger equation for real wavefunctions is reduced to solving a nonlinear first-order partial differential equation, leading to a generalization of the Prager-Hirschfelder perturbation scheme. Implications of the hydrodynamical formulation of quantum mechanics are discussed, and cases are cited where quantum and classical mechanical motions are identical.

Hirschfelder, J. O.↗

NASA NTRS1976

A strictly Markovian expansion for plasma turbulence theory

The collision operator that appears in the equation of motion for a particle distribution function that was averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. An expansion is derived for the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest nontrivial order. The validity of this expansion is seen to be the same as that of the standard quasilinear expansion.

Jones, F. C.↗

NASA NTRS1978

Three dimensional steady and unsteady asymmetric flow past wings of arbitrary planforms

The nonlinear discrete vortex method is extended to treat the problem of asymmetric flows past a wing with leading edge separation, including steady and unsteady flows. The problem is formulated in terms of a body fixed frame of reference and the nonlinear-discrete vortex method is modified accordingly. Although the method is general, only examples of flows past delta wings are presented due to the availability of experimental data as well as approximate theories. Comparison of results with experimental results for a delta wing undergoing a steady rolling motion at zero angle of attack demonstrate the superiority of the present method over existing approximate theories in obtaining highly accurate loads. Numerical results for yawed wings at large angles of attack are also presented. In all cases, total load coefficients, pressure distributions, and shapes of the free vortex sheets are shown.

Kandil, O. A.↗

NASA NTRS1978

A strictly Markovian expansion for plasma turbulence theory

The collision operator that appears in the equation of motion for a particle distribution function that has been averaged over an ensemble of random Hamiltonians is non-Markovian. It is non-Markovian in that it involves a propagated integral over the past history of the ensemble averaged distribution function. All formal expansions of this nonlinear collision operator to date preserve this non-Markovian character term by term yielding an integro-differential equation that must be converted to a diffusion equation by an additional approximation. In this note we derive an expansion of the collision operator that is strictly Markovian to any finite order and yields a diffusion equation as the lowest non-trivial order. The validity of this expansion is seen to be the same as that of the standard quasi-linear expansion.

Jones, F. C.↗

NASA NTRS1978

Stability analysis of a liquid fuel annular combustion chamber

High frequency combustion instability problems in a liquid fuel annular combustion chamber are examined. A modified Galerkin method was used to produce a set of modal amplitude equations from the general nonlinear partial differential acoustic wave equation in order to analyze the problem of instability. From these modal amplitude equations, the two variable perturbation method was used to develop a set of approximate equations of a given order of magnitude. These equations were modeled to show the effects of velocity sensitive combustion instabilities by evaluating the effects of certain parameters in the given set of equations.

Mcdonald, G. H.↗

NASA NTRS1980

Symbolic-numeric interface: A review

A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.

Ng, E. W.↗

NASA NTRS1985

Two simplified procedures for predicting cyclic material response from a strain history

Simplified inelastic analysis procedures were developed at NASA Lewis and Pratt & Whitney Aircraft for predicting the stress-strain response at the critical location of a thermomechanically cycled structure. These procedures are intended primarily for use as economical structural analysis tools in the early design stages of aircraft engine hot section components where nonlinear finite-element analyses would be prohibitively expensive. Both simplified methods use as input the total strain history calculated from a linear elastic analysis. The elastic results are modified to approximate the characteristics of the inelastic cycle by incremental solution techniques. A von Mises yield criterion is used to determine the onset of active plasticity. The fundamental assumption of these methods is that the inelastic strain is local and constrained from redistribution by the surrounding elastic material.

Kaufman, A.↗

NASA NTRS1986

A refined shear deformation theory for the analysis of laminated plates

A refined, third-order plate theory that accounts for the transverse shear strains is presented, the Navier solutions are derived for certain simply supported cross-ply and antisymmetric angle-ply laminates, and finite-element models are developed for general laminates. The new theory does not require the shear correction factors of the first-order theory (i.e., the Reissner-Mindlin plate theory) because the transverse shear stresses are represented parabolically in the present theory. A mixed finite-element model that uses independent approximations of the generalized displacements and generalized moments, and a displacement model that uses only the generalized displacements as degrees of freedom are developed. The displacement model requires C sup 1-continuity of the transverse deflection across the inter-element boundaries, whereas the mixed model requires a C sup 0-element. Also, the mixed model does not require continuous approximations (between elements) of the bending moments. Numerical results are presented to show the accuracy of the present theory in predicting the transverse stresses. Numerical results are also presented for the nonlinear bending of plates, and the results compare well with the experimental results available in the literature.

Reddy, J. N.↗

NASA NTRS1986

Nonlinear acoustic wave propagation in atmosphere

In this paper a model problem is considered that simulates an atmospheric acoustic wave propagation situation that is nonlinear. The model is derived from the basic Euler equations for the atmospheric flow and from the regular perturbations for the acoustic part. The nonlinear effects are studied by obtaining two successive linear problems in which the second one involves the solution of the first problem. Well-posedness of these problems is discussed and approximations of the radiation boundary conditions that can be used in numerical simulations are presented.

Hariharan, S. I.↗

NASA NTRS1987

Nonlinear acoustic wave propagation in atmosphere TLSP - Final report

Hariharan, S. I.↗

DOE OSTI2017

Response theory of the ergodic many-body delocalized phase: Keldysh Finkel’stein sigma models and the 10-fold way

We derive the finite temperature Keldysh response theory for interacting fermions in the presence of quenched short-ranged disorder, as applicable to any of the 10 Altland–Zirnbauer classes in an Anderson delocalized phase with at least a U(1) continuous symmetry. In this formulation of the interacting Finkel’stein nonlinear sigma model, the statistics of one-body wave functions are encoded by the constrained matrix field, while physical correlations follow from the hydrodynamic density or spin response field, which decouples the interactions. Integrating out the matrix field first, we obtain weak (anti) localization and Altshuler–Aronov quantum conductance corrections from the hydrodynamic response function. This procedure automatically incorporates the correct infrared cutoff physics, and in particular gives the Altshuler–Aronov–Khmelnitsky (AAK) equations for dephasing of weak (anti)localization due to electron–electron collisions. We explicate the method by deriving known quantumcorrections in two dimensions for the symplectic metal class AII, as well as the spin-SU(2) invariant superconductor classes C and CI. We show that quantum conductance corrections due to the special modes at zero energy in nonstandard classes are automatically cut off by temperature, as previously expected, while the Wigner–Dyson class Cooperon modes that persist to all energies are cut by dephasing. We also show that for short-ranged interactions, the standard self-consistent solution for the dephasing rate is equivalent to a particular summation of diagrams via the self-consistent Born approximation. This should be compared to the corresponding AAK solution for long-ranged Coulomb interactions, which exploits the Markovian noise correlations induced by thermal fluctuations of the electromagnetic field. We discuss prospects for exploring the many-body localization transition as a dephasing catastrophe in short-range interacting models, as encountered by approaching from the ergodic side. - Highlights: • The Keldysh response theory for disordered, interacting fermions is derived. • The theory applies to any Altland–Zirnbauer class with at least a U(1) symmetry. • The correct infrared physics at finite temperature is automatically incorporated. • The Altshuler–Aronov–Khmelnitsky equations for dephasing are systematically derived. • A strategy to explore the MBL transition as a dephasing catastrophe is proposed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2018

A new numerically stable sequential algorithm for coupled finite-strain elastoplastic geomechanics and flow

In this work, we propose a new numerically stable sequential implicit method for coupled flow and finite-strain multiplicative elastoplastic geomechanics. We find from stability analysis that the sequential method that solves the flow problem first by fixing the first Piola-Kirchhoff total stress, solving the geomechanics (solid deformation) problem at the next step, is unconditionally stable (contractive and B-stable). In this sense, this method named the fixed first Piola-Kirchhoff stress method is an extension of the fixed stress method in coupled flow and infinitesimal geomechanics. We also study the fixed second Piola-Kirchhoff stress method, comparing it with the fixed first Piola-Kirchhoff stress method, because the constitutive relations are formulated by the second Piola-Kirchhoff total stress, although fixing the second Piola-Kirchhoff total stress field does not provide theoretical unconditional stability. In space discretization, we use the finite element method for the geomechanics problem with the total Lagrangian approach, while employing the finite volume method for the flow problem. Geometrical nonlinearity from the total Lagrangian approach results in full-tensor permeability even if the initial permeability is isotropic. To consider the full-tensor permeability accurately, we employ the multipoint flux approximation in solving the flow problem. In time discretization, the backward Euler method is used. Then, we perform numerical experiments of the two fixed-stress sequential methods with various scenarios, and find superiority of their numerical stability. Specifically, the two fixed-stress sequential methods provide almost identical results for elasticity. For J2 plasticity, the two sequential methods yield slightly different numerical results although the trends of the results are similar. For the Drucker-Prager plasticity, we find that the results from the two methods are almost identical. For all the test cases, the two sequential methods are numerically stable.

58 GEOSCIENCES↗