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NASA NTRS1999

A New Method for Nonlinear and Nonstationary Time Series Analysis and Its Application to the Earthquake and Building Response Records

A new method for analyzing nonlinear and nonstationary data has been developed. The key part of the method is the Empirical Mode Decomposition method with which any complicated data set can be decomposed into a finite and often small number of Intrinsic Mode Functions (IMF). An IMF is defined as any function having the same numbers of zero-crossing and extrema, and also having symmetric envelopes defined by the local maxima and minima respectively. The IMF also admits well-behaved Hilbert transform. This decomposition method is adaptive, and, therefore, highly efficient. Since the decomposition is based on the local characteristic time scale of the data, it is applicable to nonlinear and nonstationary processes. With the Hilbert transform, the Intrinsic Mode Functions yield instantaneous frequencies as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert Spectrum, Example of application of this method to earthquake and building response will be given. The results indicate those low frequency components, totally missed by the Fourier analysis, are clearly identified by the new method. Comparisons with Wavelet and window Fourier analysis show the new method offers much better temporal and frequency resolutions.

Huang, Norden E.↗

NASA NTRS1981

Simulation of atmospheric turbulent gusts and gust gradients

An improved three-dimensional, nonrecursive model for atmospheric turbulence has been developed which provides for simulation of both instantaneous gusts and gust gradients along the flight path of the Space Shuttle. The one-dimensional gust and gust gradient spectral models, which form the basis for the simulation scheme, are developed from three-dimensional, von Karman spectra, integrated over finite limits based on the characteristic dimensions (length, width, and thickness) of the flight vehicle. By means of the simulation process nondimensional time series for both gusts and gust gradients have been generated and stored on a series of magnetic tapes for four altitude bands ranging from 0 to 10,000 meters. These Shuttle Simulated Turbulence Tapes (SSTT) have been validated by means of spectral and statistical analyses with satisfactory results. The dimensionless form of the time series coupled with the use of a von Karman spectral model (as opposed to a Dryden model) distinguish the SSTT from earlier turbulence simulation concepts.

Tatom, F. B.↗

NASA NTRS1992

Validation of three-dimensional incompressible spatial direct numerical simulation code: A comparison with linear stability and parabolic stability equation theories for boundary-layer transition on a flat plate

Spatially evolving instabilities in a boundary layer on a flat plate are computed by direct numerical simulation (DNS) of the incompressible Navier-Stokes equations. In a truncated physical domain, a nonstaggered mesh is used for the grid. A Chebyshev-collocation method is used normal to the wall; finite difference and compact difference methods are used in the streamwise direction; and a Fourier series is used in the spanwise direction. For time stepping, implicit Crank-Nicolson and explicit Runge-Kutta schemes are used to the time-splitting method. The influence-matrix technique is used to solve the pressure equation. At the outflow boundary, the buffer-domain technique is used to prevent convective wave reflection or upstream propagation of information from the boundary. Results of the DNS are compared with those from both linear stability theory (LST) and parabolized stability equation (PSE) theory. Computed disturbance amplitudes and phases are in very good agreement with those of LST (for small inflow disturbance amplitudes). A measure of the sensitivity of the inflow condition is demonstrated with both LST and PSE theory used to approximate inflows. Although the DNS numerics are very different than those of PSE theory, the results are in good agreement. A small discrepancy in the results that does occur is likely a result of the variation in PSE boundary condition treatment in the far field. Finally, a small-amplitude wave triad is forced at the inflow, and simulation results are compared with those of LST. Again, very good agreement is found between DNS and LST results for the 3-D simulations, the implication being that the disturbance amplitudes are sufficiently small that nonlinear interactions are negligible.

Joslin, Ronald D.↗

NASA NTRS1974

The modified equation approach to the stability and accuracy analysis of finite-difference methods

The stability and accuracy of finite-difference approximations to simple linear partial differential equations are analyzed by studying the modified partial differential equation. Aside from round-off error, the modified equation represents the actual partial differential equation solved when a numerical solution is computed using a finite-difference equation. The modified equation is derived by first expanding each term of a difference scheme in a Taylor series and then eliminating time derivatives higher than first order by certain algebraic manipulations. The connection between 'heuristic' stability theory based on the modified equation approach and the von Neumann (Fourier) method is established. In addition to the determination of necessary and sufficient conditions for computational stability, a truncated version of the modified equation can be used to gain insight into the nature of both dissipative and dispersive errors.

Warming, R. F.↗

NASA NTRS2005

Physics-based Simulation of Human Posture Using 3D Whole Body Scanning Technology for Astronaut Space Suit Evaluation

Over the past few years high precision three-dimensional (3D) full body laser scanners have been developed to be used as a powerful anthropometry tool for quantification of the morphology of the human body. The full body scanner can quickly extract body characteristics in non-contact fashion. It is required for the Anthropometry and Biomechanics Facility (ABF) to have capabilities for kinematics simulation of a digital human at various postures whereas the laser scanner only allows capturing a single static posture at each time. During this summer fellowship period a theoretical study has been conducted to estimate an arbitrary posture with a series of example postures through finite element (FE) approximation and found that four-point isoparametric FE approximation would result in reasonable maximum position errors less than 5%. Subsequent pilot scan experiments demonstrated that a bead marker with a nominal size of 6 mm could be used as a marker for digitizing 3-D coordinates of anatomical landmarks for further kinematic analysis. Two sessions of human subject testing were conducted for reconstruction of an arbitrary postures from a set of example postures for each joint motion for the forearm/hand complex and the whole upper extremity.

Kim, Kyu-Jung↗

DOE OSTI2025

Numerical schemes for 3-wave kinetic equations: A complete treatment of the collision operator

In our previous work Walton and Tran (2023), numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the long time dynamics of the equation given the multiple blow-up time phenomenon. In this second work in the series, we propose numerical treatments for the complete 3-wave kinetic equations, in which the complete, much more complicated collision operators are fully considered based on a novel conservative form of the equation. Here we then derive an implicit finite volume scheme to solve the equation. The new discretization uses an adaptive time-stepping method which allows for the simulations to be carried to very long times. Our computed solutions are compared with previously derived long-time asymptotic estimates for the decay rate of total energy of time-dependent solutions of 3-wave kinetic equations and found to be in excellent agreement.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Time-Resolved Stochastic Dynamics of Quantum Thermal Machines

Steady-state quantum thermal machines are typically characterized by a continuous flow of heat between different reservoirs. However, at the level of discrete stochastic realizations, heat flow is unraveled as a series of abrupt quantum jumps, each representing an exchange of finite quanta with the environment. Here, in this work, we present a framework that resolves the dynamics of quantum thermal machines into cycles classified as enginelike, coolinglike, or idle. We analyze the statistics of individual cycle types and their durations, enabling us to determine both the fraction of cycles useful for thermodynamic tasks and the average waiting time between cycles of a given type. Central to our analysis is the notion of intermittency, which captures the operational consistency of the machine by assessing the frequency and distribution of idle cycles. Our framework offers a novel approach to characterizing thermal machines, with significant relevance to experiments involving mesoscopic transport through quantum dots.

full counting statistics↗

NASA NTRSDate not supplied

Methodology and Development of SLS Ascent Loads

The methodology and development of the Space Launch System (SLS) Block 1 Boost and Core phase Ascent loads will be presented in this paper. As is common practice, a series of fixed time/mass Finite Element Models of the vehicle were developed for the purpose of evaluating the structural loads and accelerations for particular Mach or time ranges associated with the Boost and Core phases of flight. Load contributors, which vary between bins, include the flight mechanic induced portion of the loads (e.g., Static-Elastic (STEL) or Vehicle Load Indicator (VLI) tool results), gust loads, buffet loads, Solid Rocket Booster thrust oscillations, Core Stage Engine thrust oscillations, maneuvering loads, thrust vector-related dispersions, and Programmed Test Input induced loads. Each of these load contributors will be examined in this paper, as well as the Loads Combination Equations used to generate the final loads. In addition to the typical coupled loads analysis, an Ascent loads evaluation is included in the SLS Artemis 1 Day of Launch process; a summary of this evaluation will also be presented in this paper.

SLS↗

NASA NTRS2003

Update on EMD and Hilbert-Spectra Analysis of Time Series

This method is especially well suited for analyzing time-series data that represent nonstationary and nonlinear physical phenomena. The method is based principally on the concept of empirical mode decomposition (EMD), according to which any complicated signal (as represented by digital samples) can be decomposed into a finite number of functions, called "intrinsic mode functions" (IMFs), that admit well-behaved Hilbert transforms. The local energies and the instantaneous frequencies derived from the IMFs through Hilbert transforms can be used to construct an energy-frequency-time distribution, denoted a Hilbert spectrum.

Huang, Norden E.↗

NASA NTRS2005

Application of the Hilbert-Huang Transform to Financial Data

A paper discusses the application of the Hilbert-Huang transform (HHT) method to time-series financial-market data. The method was described, variously without and with the HHT name, in several prior NASA Tech Briefs articles and supporting documents. To recapitulate: The method is especially suitable for analyzing time-series data that represent nonstationary and nonlinear phenomena including physical phenomena and, in the present case, financial-market processes. The method involves the empirical mode decomposition (EMD), in which a complicated signal is decomposed into a finite number of functions, called "intrinsic mode functions" (IMFs), that admit well-behaved Hilbert transforms. The HHT consists of the combination of EMD and Hilbert spectral analysis. The local energies and the instantaneous frequencies derived from the IMFs through Hilbert transforms can be used to construct an energy-frequency-time distribution, denoted a Hilbert spectrum. The instant paper begins with a discussion of prior approaches to quantification of market volatility, summarizes the HHT method, then describes the application of the method in performing time-frequency analysis of mortgage-market data from the years 1972 through 2000. Filtering by use of the EMD is shown to be useful for quantifying market volatility.

Huang, Norden↗

NASA NTRS1976

An algorithm for the weighting matrices in the sampled-data optimal linear regulator problem

The sampled-data optimal linear regulator problem provides a means whereby a control designer can use an understanding of continuous optimal regulator design to produce a digital state variable feedback control law which satisfies continuous system performance specifications. A basic difficulty in applying the sampled-data regulator theory is the requirement that certain digital performance index weighting matrices, expressed as complicated functions of system matrices, be computed. Infinite series representations are presented for the weighting matrices of the time-invariant version of the optimal linear sampled-data regulator problem. Error bounds are given for estimating the effect of truncating the series expressions after a finite number of terms, and a method is described for their computer implementation. A numerical example is given to illustrate the results.

Armstrong, E. S.↗

NASA NTRS2003

Acoustical Applications of the HHT Method

A document discusses applications of a method based on the Huang-Hilbert transform (HHT). The method was described, without the HHT name, in Analyzing Time Series Using EMD and Hilbert Spectra (GSC-13817), NASA Tech Briefs, Vol. 24, No. 10 (October 2000), page 63. To recapitulate: The method is especially suitable for analyzing time-series data that represent nonstationary and nonlinear physical phenomena. The method involves the empirical mode decomposition (EMD), in which a complicated signal is decomposed into a finite number of functions, called intrinsic mode functions (IMFs), that admit well-behaved Hilbert transforms. The HHT consists of the combination of EMD and Hilbert spectral analysis.

Huang, Norden E.↗

DOE OSTI2024

Measuring the Loschmidt Amplitude for Finite-Energy Properties of the Fermi-Hubbard Model on an Ion-Trap Quantum Computer

Calculating the equilibrium properties of condensed-matter systems is one of the promising applications of near-term quantum computing. Recently, hybrid quantum-classical time-series algorithms have been proposed to efficiently extract these properties from a measurement of the Loschmidt amplitude ⟨ ψ | e − i H ^ t | ψ ⟩ from initial states | ψ ⟩ and a time evolution under the Hamiltonian H ^ up to short times t . In this work, we study the operation of this algorithm on a present-day quantum computer. Specifically, we measure the Loschmidt amplitude for the Fermi-Hubbard model on a 16 -site ladder geometry (32 orbitals) on the Quantinuum H2-1 trapped-ion device. We assess the effect of noise on the Loschmidt amplitude and implement algorithm-specific error-mitigation techniques. By using a thus-motivated error model, we numerically analyze the influence of noise on the full operation of the quantum-classical algorithm by measuring expectation values of local observables at finite energies. Finally, we estimate the resources needed for scaling up the algorithm. Published by the American Physical Society 2024

Physics↗

NASA NTRS1993

Blasim: A computational tool to assess ice impact damage on engine blades

A portable computer called BLASIM was developed at NASA LeRC to assess ice impact damage on aircraft engine blades. In addition to ice impact analyses, the code also contains static, dynamic, resonance margin, and supersonic flutter analysis capabilities. Solid, hollow, superhybrid, and composite blades are supported. An optional preprocessor (input generator) was also developed to interactively generate input for BLASIM. The blade geometry can be defined using a series of airfoils at discrete input stations or by a finite element grid. The code employs a coarse, fixed finite element mesh containing triangular plate finite elements to minimize program execution time. Ice piece is modeled using an equivalent spherical objective that has a high velocity opposite that of the aircraft and parallel to the engine axis. For local impact damage assessment, the impact load is considered as a distributed force acting over a region around the impact point. The average radial strain of the finite elements along the leading edge is used as a measure of the local damage. To estimate damage at the blade root, the impact is treated as an impulse and a combined stress failure criteria is employed. Parametric studies of local and root ice impact damage, and post-impact dynamics are discussed for solid and composite blades.

Reddy, E. S.↗

NASA NTRS1993

BLASIM - A computational tool to assess ice impact damage on engine blades

A portable computer code called BLASIM is developed at NASA LeRC to assess the ice impact damage on aircraft engine blades. In addition to the ice impact analyses, the code is also capable of carrying out static, dynamic, resonance margin and flutter analyses. The blade can be solid, hollow, superhybrid or composite material. An optional preprocessor (input generator) is also developed to generate input to the code through interactive process. The blade geometry can be defined either by a series of airfoils at discrete input stations or by a finite element grid. The code employs a coarse fixed finite element mesh with triangular plate finite elements and has quick turnaround time. The ice piece is modeled as an equivalent spherical object and has the velocity opposite to that of the aircraft with direction parallel to the engine axis. For the local impact damage assessment, the impact force is considered as a distributed load acting over a region around the impact point and the average radial strain of the finite elements along the leading edge is taken as a measure of the local damage. To estimate the damage at the blade root, the impact is considered to be an impulse and a combined stress failure criteria is employed. Parametric studies for local and root ice impact damage, and post-impact dynamics are discussed for solid and composite blades.

Reddy, E. S.↗

NASA NTRSDate not supplied

High Speed Data Investigation of an LOX/LNG Combustor Assisted by Non-linear Acoustic Simulations

During testing of a regenerative cooled liquid oxygen/liquid natural gas combustor large amplitude combustion instabilities were observed in half of the series conducted. These instabilities produced an unsteady pressure Root Mean Square (RMS) value greater than the chamber steady state pressure. Quick look investigations of the test data presented some standard indications of non-linear tangential instability such as a non-sinusoidal time series, and larger peaks than trough. Also observed in the data quick look were some atypical findings including a first harmonic amplitude larger than the fundamental, and numerous side bands on the fundamental and subsequent harmonics. A simplified model of the combustor was created using a commercially available finite element software. The first tangential acoustic mode was then simulated using a non-linear wave equation in conjunction with a time explicit discontinuous Gelarkin flow solver. Numerous simulations were evaluated providing valuable insight and reproducing the atypical findings in the data. Side bands were created through non-linear signal modulation like an AM radio. This happens due to the non-linear steepening and relaxing that occurs in large amplitude oscillations. The simulation was also able to reproduce a signal with a first harmonic amplitude larger than the fundamental. Investigating the pressure signal at multiple locations across the chamber head end revealed radial dependent frequency content. A literature review of the topic revealed corroboration of these effect in idealized geometries.

Combustion Stability↗

NASA NTRSDate not supplied

High Speed Data Investigation of an LOX/LNG Combustor Assisted by Non-linear Acoustic Simulations

Combustion Stability↗

NASA NTRS1991

Shedding patterns of the near-wake vortices behind a circular cylinder

The unsteady incompressible Navier-Stokes equations have been accurately solved for the laminar flow past a circular cylinder in the Reynolds number range 50-200. A direct elliptic solver called the SEVP is used to rapidly advance the streamfunction in time, facilitating the overall convergence to the fully periodic or quasi-steady state. A new integral-series method is developed for the far-field streamfunction condition on a finite two-dimensional computational domain. The use of fourth-order Hermitain relations for the convection terms in the conservation-form vorticity transport equation has also contributed to the good comparison of the present results with the earlier experimental data. The vortex-shedding patterns visualized by the experimentalist are numerically reproduced here in the given Reynolds number range. Discussions that may be helpful in interpreting the behavior of the shedding frequency are presented in the main text.

Sa, Jong-Youb↗