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DOE OSTI2020

Numerically modeling stochastic inflation in slow-roll and beyond

In this work, we present a complete numerical treatment of inflationary dynamics under the influence of stochastic corrections from sub-Hubble modes. We discuss how to exactly model the stochastic noise terms arising from the sub-Hubble quantum modes that give rise to the coarse-grained inflaton dynamics in the form of stochastic differential equations. The stochastic differential equations are solved event by event on a discrete time grid. We then compute the power spectrum of curvature perturbations that can be compared with the power spectrum computed in the traditional fashion using the Mukhanov-Sasaki equation by canonically quantizing the inflaton fluctuations. Our numerical procedure helps us to easily extend the formalism to ultra slow-roll inflation and study the possibility of primordial black hole formation.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1982

Flap-lag-torsional dynamics of extensional and inextensional rotor blades in hover and in forward flight

The reduction of the O(cu epsilon) integro differential equations to ordinary differential equations using a set of orthogonal functions is described. Attention was focused on the hover flight condition. The set of Galerkin integrals that appear in the reduced equations was evaluated by making use of nonrotating beam modes. Although a large amount of computer time was needed to accomplish this task, the Galerkin integrals so evaluated were stored on tape on a permanent basis. Several of the coefficients were also obtained in closed form in order to check the accuracy of the numerical computations. The equilibrium solution to the set of 3n equations obtained was determined as the solution to a minimization problem.

Dasilva, C.↗

NASA NTRS1985

Topics in spectral methods

After detailing the construction of spectral approximations to time-dependent mixed initial boundary value problems, a study is conducted of differential equations of the form 'partial derivative of u/partial derivative of t = Lu + f', where for each t, u(t) belongs to a Hilbert space such that u satisfies homogeneous boundary conditions. For the sake of simplicity, it is assumed that L is an unbounded, time-independent linear operator. Attention is given to Fourier methods of both Galerkin and pseudospectral method types, the Galerkin method, the pseudospectral Chebyshev and Legendre methods, the error equation, hyperbolic partial differentiation equations, and time discretization and iterative methods.

Gottlieb, D.↗

NASA NTRS1972

Dynamic characteristics of a two-stage variable-mass flexible missile with internal flow

A general formulation of the dynamical problems associated with powered flight of a two stage flexible, variable-mass missile with internal flow, discrete masses, and aerodynamic forces is presented. The formulation comprises six ordinary differential equations for the rigid body motion, 3n ordinary differential equations for the n discrete masses and three partial differential equations with the appropriate boundary conditions for the elastic motion. This set of equations is modified to represent a single stage flexible, variable-mass missile with internal flow and aerodynamic forces. The rigid-body motion consists then of three translations and three rotations, whereas the elastic motion is defined by one longitudinal and two flexural displacements, the latter about two orthogonal transverse axes. The differential equations are nonlinear and, in addition, they possess time-dependent coefficients due to the mass variation.

Meirovitch, L.↗

NASA NTRS1972

Dissipative systems.

A theory is presented that shows how the concept of dissipative systems of ordinary differential equations can be extended to include a broad class of functional and partial differential equations, such as retarded functional differential equations and parabolic partial differential equations. Since the basic hypotheses are all in terms of boundedness, finding sufficient conditions in terms of Liapunov functions would not be too difficult.

Lasalle, J. P.↗

NASA NTRS1993

Multiple large-scale coherent mode interactions in a developing round jet

The integral energy method has been used to study the nonlinear interactions of the large-scale coherent structure in a spatially developing round jet. The streamwise development of a jet is obtained in terms of the mean flow shear-layer momentum thickness, the wave-mode kinetic energy and the wave-mode phase angle. With the energy method, a system of partial differential equations is reduced to a system of ordinary differential equations. The nonlinear differential equations are solved with initial conditions which are given at the nozzle exit. It is shown that the initial wave-mode energy densities as well as the initial phase angles play a significant role in the streamwise evolution of the large-scale coherent wave modes and the mean flow.

Lee, Sang S.↗

DOE OSTI2020

A scalable matrix-free spectral element approach for unsteady PDE constrained optimization using PETSc/TAO

In this work, we provide a new approach for the efficient matrix-free application of the transpose of the Jacobian for the spectral element method for the adjoint-based solution of partial differential equation (PDE) constrained optimization. This results in optimizations of nonlinear PDEs using explicit integrators where the integration of the adjoint problem is not more expensive than the forward simulation. Solving PDE constrained optimization problems entails combining expertise from multiple areas, including simulation, computation of derivatives, and optimization. The Portable, Extensible Toolkit for Scientific computation (PETSc) together with its companion package, the Toolkit for Advanced Optimization (TAO), is an integrated numerical software library that contains an algorithmic/software stack for solving linear systems, nonlinear systems, ordinary differential equations, differential algebraic equations, and large-scale optimization problems and, as such, is an ideal tool for performing PDE-constrained optimization. This paper describes an efficient approach in which the software stack provided by PETSc/TAO can be used for large-scale nonlinear time-dependent problems. Time integration can involve a range of high-order methods, both implicit and explicit. The PDE-constrained optimization algorithm used is gradient-based and seamlessly integrated with the simulation of the physical problem.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2008

Oxidation Behavior of Carbon Fiber-Reinforced Composites

OXIMAP is a numerical (FEA-based) solution tool capable of calculating the carbon fiber and fiber coating oxidation patterns within any arbitrarily shaped carbon silicon carbide composite structure as a function of time, temperature, and the environmental oxygen partial pressure. The mathematical formulation is derived from the mechanics of the flow of ideal gases through a chemically reacting, porous solid. The result of the formulation is a set of two coupled, non-linear differential equations written in terms of the oxidant and oxide partial pressures. The differential equations are solved simultaneously to obtain the partial vapor pressures of the oxidant and oxides as a function of the spatial location and time. The local rate of carbon oxidation is determined at each time step using the map of the local oxidant partial vapor pressure along with the Arrhenius rate equation. The non-linear differential equations are cast into matrix equations by applying the Bubnov-Galerkin weighted residual finite element method, allowing for the solution of the differential equations numerically.

Sullivan, Roy M.↗

DOE OSTI2024

Solving differential‐algebraic equations in power system dynamic analysis with quantum computing

Abstract Power system dynamics are generally modeled by high dimensional non‐linear differential‐algebraic equations (DAEs) given a large number of components forming the network. These DAEs' complexity can grow exponentially due to the increasing penetration of distributed energy resources, whereas their computation time becomes sensitive due to the increasing interconnection of the power grid with other energy systems. This paper demonstrates the use of quantum computing algorithms to solve DAEs for power system dynamic analysis. We leverage a symbolic programming framework to equivalently convert the power system's DAEs into ordinary differential equations (ODEs) using index reduction methods and then encode their data into qubits using amplitude encoding. The system non‐linearity is captured by Hamiltonian simulation with truncated Taylor expansion so that state variables can be updated by a quantum linear equation solver. Our results show that quantum computing can solve the power system's DAEs accurately with a computational complexity polynomial in the logarithm of the system dimension. We also illustrate the use of recent advanced tools in scientific machine learning for implementing complex computing concepts, that is, Taylor expansion, DAEs/ODEs transformation, and quantum computing solver with abstract representation for power engineering applications.

computational complexity↗

NASA NTRS2002

Parametric Identification of Nonlinear Dynamical Systems

In this project, we looked at the application of harmonic balancing as a tool for identifying parameters (HBID) in a nonlinear dynamical systems with chaotic responses. The main idea is to balance the harmonics of periodic orbits extracted from measurements of each coordinate during a chaotic response. The periodic orbits are taken to be approximate solutions to the differential equations that model the system, the form of the differential equations being known, but with unknown parameters to be identified. Below we summarize the main points addressed in this work. The details of the work are attached as drafts of papers, and a thesis, in the appendix. Our study involved the following three parts: (1) Application of the harmonic balance to a simulation case in which the differential equation model has known form for its nonlinear terms, in contrast to a differential equation model which has either power series or interpolating functions to represent the nonlinear terms. We chose a pendulum, which has sinusoidal nonlinearities; (2) Application of the harmonic balance to an experimental system with known nonlinear forms. We chose a double pendulum, for which chaotic response were easily generated. Thus we confronted a two-degree-of-freedom system, which brought forth challenging issues; (3) A study of alternative reconstruction methods. The reconstruction of the phase space is necessary for the extraction of periodic orbits from the chaotic responses, which is needed in this work. Also, characterization of a nonlinear system is done in the reconstructed phase space. Such characterizations are needed to compare models with experiments. Finally, some nonlinear prediction methods can be applied in the reconstructed phase space. We developed two reconstruction methods that may be considered if the common method (method of delays) is not applicable.

Feeny, Brian↗

NASA NTRS1976

The method of Ritz applied to the equation of Hamilton

Without any reference to the theory of differential equations, the initial value problem of the nonlinear, nonconservative double pendulum system is solved by the application of the method of Ritz to the equation of Hamilton. Also shown is an example of the reduction of the traditional eigenvalue problem of linear, homogeneous, differential equations of motion to the solution of a set of nonhomogeneous algebraic equations. No theory of differential equations is used. Solution of the time-space path of the linear oscillator is demonstrated and compared to the exact solution.

Bailey, C. D.↗

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↗

NASA NTRS2003

Preliminary Planar Formation: Flight Dynamics Near Sun-Earth L2 Point

NASA's Goddard Space Flight Center is planning a series of missions in the vicinity of the Sun-Earth L2 libration point. Some of these projects will involve a distributed space system of telescope spacecraft acting together as a single telescope for high-resolution. The individual telescopes will be configured in a plane, surrounding a hub, where the telescope plane can be aimed toward various astronomical targets of interest. In preparation for these missions, it is necessary to develop an improved understanding of the dynamical behavior of objects in a planar configuration near L2. The classical circular restricted three body problem is taken as the basis for the analysis. At first order, the motion of such a telescope relative to the hub is described by a system of linear second order differential equations. These equations are identical to the circular restricted problem's linear equations describing the hub motion about L2. Therefore, the fundamental frequencies, both parallel to and normal to the ecliptic plane, are the same for the relative telescope motion as for the hub motion. To maintain the telescope plane for the duration necessary for the planned observations, a halo-type orbit of the telescopes about the hub is investigated. By using a halo orbit, the individual telescopes remain in approximately the same plane over the observation duration. For such an orbit, the fundamental periods parallel to and normal to the ecliptic plane are forced to be the same by careful selection of the initial conditions in order to adjust the higher order forces. The relative amplitudes of the resulting oscillations are associated with the orientation of the telescope plane relative to the ecliptic. As in the circular restricted problem, initial conditions for the linearized equations must be selected so as not to excite the convergent or divergent linear modes. In a higher order analysis, the telescope relative motion equations include the effects of the position of the hub relative to L2. In this paper, the differential equations are developed through second order in the distance of the hub from the libration point. A modified Lindstedt-Poincad perturbation method is employed to construct the solution of these differential equations through that same order of magnitude. In the course of the solution process, relationships are determined between the initial conditions of the telescopes, selected in order to avoid resonance excitation. As the differential equations include the hub position, it is necessary to simultaneously develop the solution for the hub. As has been done in past analyses of the circular restricted problem, the hub position is written in a power series formulation in terms of its distance from L2. Then, in order to be included in the telescope equations, the hub solution is cast in terms of the nonlinear frequency of the relative telescope motion. In the course of the analysis, it is determined that the hub should also maintain a halo orbit - about L2. Additionally, relationships are formed between the initial conditions of the telescopes and the hub. These relationships may be used to associate sets of initial conditions with particular orientations of the telescope plane. The accuracy of the analytical solution is verified through various simulations and comparison to numerical integration of the differential equations. The results of the simulations are presented, along with a graphical representation of the relationships between the initial conditions of the telescopes and hub.

Segerman, Alan M.↗

NASA NTRS1976

Nonlinear Equations of Motion for Cantilever Rotor Blades in Hover with Pitch Link Flexibility, Twist, Precone, Droop, Sweep, Torque Offset, and Blade Root Offset

Nonlinear equations of motion for a cantilever rotor blade are derived for the hovering flight condition. The blade is assumed to have twist, precone, droop, sweep, torque offset and blade root offset, and the elastic axis and the axes of center of mass, tension, and aerodynamic center coincident at the quarter chord. The blade is cantilevered in bending, but has a torsional root spring to simulate pitch link flexibility. Aerodynamic forces acting on the blade are derived from strip theory based on quasi-steady two-dimensional airfoil theory. The equations are hybrid, consisting of one integro-differential equation for root torsion and three integro-partial differential equations for flatwise and chordwise bending and elastic torsion. The equations are specialized for a uniform blade and reduced to nonlinear ordinary differential equations by Galerkin's method. They are linearized for small perturbation motions about the equilibrium operating condition. Modal analysis leads to formulation of a standard eigenvalue problem where the elements of the stability matrix depend on the solution of the equilibrium equations. Two different forms of the root torsion equation are derived that yield virtually identical numerical results. This provides a reasonable check for the accuracy of the equations.

Hodges, D. H.↗

DOE OSTI2024

BPZ equations for higher degenerate fields and nonperturbative Dyson-Schwinger equations

In the two-dimensional Liouville conformal field theory, correlation functions involving a degenerate field satisfy partial differential equations due to the decoupling of the null descendant field. On the other hand, the instanton partition function of a four-dimensional N = 2 supersymmetric theory in the Ω -background at a special point of the parameter space also satisfies a partial differential equation resulting from the constraints of the gauge field configurations. This partial differential equation can be proved using the nonperturbative Dyson-Schwinger equations. We show for the next-to-simplest case that the partial differential equations obtained from two different perspectives can be identified, thereby confirming an assertion of the Bogomol’nyi-Prasad-Sommerfield/conformal field theory correspondence. Published by the American Physical Society 2024

Astronomy & Astrophysics↗

NASA NTRS1984

Analytical results for postbuckling behavior of plates in compression and shear

The postbuckling behavior of long rectangular isotropic and orthotropic plates is determined. By assuming trigonometric functions in one direction, the nonlinear partial differential equations of von Karman large deflection plate theory are converted into nonlinear ordinary differential equations. The ordinary differential equations are solved numerically using an available boundary value problem solver which makes use of Newton's method. Results for longitudinal compression show different postbuckling behavior between isotropic and orthotropic plates. Results for shear show that change in inplane edge constraints can cause large change in postbuckling stiffness.

Stein, M.↗

NASA NTRS1985

Analytical results for post-buckling behaviour of plates in compression and in shear

Stein, M.↗

DOE OSTI2021

Vibration-Based Sensor Design: A Grey-Box Approach

Knowledge of the internal structure of an object or device under investigation proceeds from the basic idea of constructing its dynamic behavioral relations governed by a set of differential/algebraic equations that characterize its response. These equations can be partial differential equations leading to finite element or finite difference relations requiring a complex numerical solution on a super computer or ordinary differential equations requiring sophisticated numerical integration techniques to obtain the desired solution. Discrete dynamic systems evolving from digitized data acquisition are typically captured by sampled-data (continuous-to-discrete) representations characterized by a set of difference equations specifying the underlying system dynamics. In any case, with a mathematical description in hand, Grey-Box modeling techniques have evolved, concerned with the estimation of model parameters embedded in a prescribed set of equations (the system) governing its behavior, while capturing the underlying physical phenomenology of the problem at hand.

97 MATHEMATICS AND COMPUTING↗