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NASA NTRS1995

Absorbing Boundary Conditions For Optical Pulses In Dispersive, Nonlinear Materials

This paper will present results in computational nonlinear optics. An algorithm will be described that provides absorbing boundary conditions for optical pulses in dispersive, nonlinear materials. A new numerical absorber at the boundaries has been developed that is responsive to the spectral content of the pulse. Also, results will be shown of calculations of 2-D electromagnetic nonlinear waves computed by directly integrating in time the nonlinear vector Maxwell's equations. The results will include simulations of "light bullet" like pulses. Here diffraction and dispersion will be counteracted by nonlinear effects. Comparisons will be shown of calculations that use the standard boundary conditions and the new ones.

Goorjian, Peter M.↗

NASA NTRS2001

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

This chapter describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.↗

NASA NTRS2002

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

This talk describes some of the building blocks to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.↗

NASA NTRS2002

A Nonlinear, Human-Centered Approach to Motion Cueing with a Neurocomputing Solver

This paper discusses the continuation of research into the development of new motion cueing algorithms first reported in 1999. In this earlier work, two viable approaches to motion cueing were identified: the coordinated adaptive washout algorithm or 'adaptive algorithm', and the 'optimal algorithm'. In this study, a novel approach to motion cueing is discussed that would combine features of both algorithms. The new algorithm is formulated as a linear optimal control problem, incorporating improved vestibular models and an integrated visual-vestibular motion perception model previously reported. A control law is generated from the motion platform states, resulting in a set of nonlinear cueing filters. The time-varying control law requires the matrix Riccati equation to be solved in real time. Therefore, in order to meet the real time requirement, a neurocomputing approach is used to solve this computationally challenging problem. Single degree-of-freedom responses for the nonlinear algorithm were generated and compared to the adaptive and optimal algorithms. Results for the heave mode show the nonlinear algorithm producing a motion cue with a time-varying washout, sustaining small cues for a longer duration and washing out larger cues more quickly. The addition of the optokinetic influence from the integrated perception model was shown to improve the response to a surge input, producing a specific force response with no steady-state washout. Improved cues are also observed for responses to a sway input. Yaw mode responses reveal that the nonlinear algorithm improves the motion cues by reducing the magnitude of negative cues. The effectiveness of the nonlinear algorithm as compared to the adaptive and linear optimal algorithms will be evaluated on a motion platform, the NASA Langley Research Center Visual Motion Simulator (VMS), and ultimately the Cockpit Motion Facility (CMF) with a series of pilot controlled maneuvers. A proposed experimental procedure is discussed. The results of this evaluation will be used to assess motion cueing performance.

Telban, Robert J.↗

NASA NTRS2002

Nonlinear Dynamic Models in Advanced Life Support

To facilitate analysis, ALS systems are often assumed to be linear and time invariant, but they usually have important nonlinear and dynamic aspects. Nonlinear dynamic behavior can be caused by time varying inputs, changes in system parameters, nonlinear system functions, closed loop feedback delays, and limits on buffer storage or processing rates. Dynamic models are usually cataloged according to the number of state variables. The simplest dynamic models are linear, using only integration, multiplication, addition, and subtraction of the state variables. A general linear model with only two state variables can produce all the possible dynamic behavior of linear systems with many state variables, including stability, oscillation, or exponential growth and decay. Linear systems can be described using mathematical analysis. Nonlinear dynamics can be fully explored only by computer simulations of models. Unexpected behavior is produced by simple models having only two or three state variables with simple mathematical relations between them. Closed loop feedback delays are a major source of system instability. Exceeding limits on buffer storage or processing rates forces systems to change operating mode. Different equilibrium points may be reached from different initial conditions. Instead of one stable equilibrium point, the system may have several equilibrium points, oscillate at different frequencies, or even behave chaotically, depending on the system inputs and initial conditions. The frequency spectrum of an output oscillation may contain harmonics and the sums and differences of input frequencies, but it may also contain a stable limit cycle oscillation not related to input frequencies. We must investigate the nonlinear dynamic aspects of advanced life support systems to understand and counter undesirable behavior.

Jones, Harry↗

NASA NTRS2003

Nonlinear Observers for Gyro Calibration

High precision estimation and control algorithms, to achieve unprecedented levels of pointing accuracy, will be required to support future formation flying missions such as interferometry missions. Achieving high pointing accuracy requires precise knowledge of the spacecraft rotation rate. Typically, the rotation rate is measured by a gyro. The measured rates can be corrupted by errors in alignment and scale factor, gyro biases, and noise. In this work, we present nonlinear observers for gyro calibration. Nonlinear observers are superior to extended or pseudo-linear Kalman filter type approaches for large errors and global stability. Three nonlinear gyro calibration observers are developed. The first observer estimates a constant gyro bias. The second observer estimates scale factor errors. The third observer estimates the gyro alignment for three orthogonal gyros. The convergence properties of all three observers are discussed. Additionally, all three observers are coupled with a nonlinear control algorithm. The stability of each of the resulting closed loop systems is analyzed. The observers are then combined, and the gyro calibration parameters are estimated simultaneously. The stability of the combined observers is addressed, as well as the stability of the resulting closed loop systems. Simulated test results are presented for each scenario. Finally, the nonlinear observers are compared to a pseudo-linear Kalman filter.

Thienel, Julie↗

NASA NTRS2004

Reliability of Complex Nonlinear Numerical Simulations

This work describes some of the procedure to ensure a higher level of confidence in the predictability and reliability (PAR) of numerical simulation of multiscale complex nonlinear problems. The focus is on relating PAR of numerical simulations with complex nonlinear phenomena of numerics. To isolate sources of numerical uncertainties, the possible discrepancy between the chosen partial differential equation (PDE) model and the real physics and/or experimental data is set aside. The discussion is restricted to how well numerical schemes can mimic the solution behavior of the underlying PDE model for finite time steps and grid spacings. The situation is complicated by the fact that the available theory for the understanding of nonlinear behavior of numerics is not at a stage to fully analyze the nonlinear Euler and Navier-Stokes equations. The discussion is based on the knowledge gained for nonlinear model problems with known analytical solutions to identify and explain the possible sources and remedies of numerical uncertainties in practical computations. Examples relevant to turbulent flow computations are included.

Yee, H. C.↗

NASA NTRS2005

Nonlinear Reduced Order Random Response Analysis of Structures With Shallow Curvature

The goal of this investigation is to further develop nonlinear modal numerical simulation methods for application to geometrically nonlinear response of structures with shallow curvature under random loadings. For reduced order analysis, the modal basis selection must be capable of reflecting the coupling in both the linear and nonlinear stiffness. For the symmetric shallow arch under consideration, four categories of modal basis functions are defined. Those having symmetric transverse displacements (ST modes) can be designated as transverse dominated (ST-T) modes and in-plane dominated (ST-I) modes. Those having anti-symmetric transverse displacements (AT modes) can similarly be designated as transverse dominated (AT-T) modes and in-plane dominated (AT-I) modes. The response of an aluminum arch under a uniformly distributed transverse random loading is investigated. Results from nonlinear modal simulations made using various modal bases are compared with those obtained from a numerical simulation in physical degrees-of-freedom. While inclusion of ST-T modes is important for all response regimes, it is found that the ST-I modes become increasingly important in the nonlinear response regime, and that AT-T and AT-I modes are critical in the autoparametric regime.

Przekop, Adam↗

NASA NTRS2005

Building Blocks for Reliable Complex Nonlinear Numerical Simulations

Yee, H. C.↗

NASA NTRS2005

Recent Applications of Higher-Order Spectral Analysis to Nonlinear Aeroelastic Phenomena

Recent applications of higher-order spectral (HOS) methods to nonlinear aeroelastic phenomena are presented. Applications include the analysis of data from a simulated nonlinear pitch and plunge apparatus and from F-18 flight flutter tests. A MATLAB model of the Texas A&MUniversity s Nonlinear Aeroelastic Testbed Apparatus (NATA) is used to generate aeroelastic transients at various conditions including limit cycle oscillations (LCO). The Gaussian or non-Gaussian nature of the transients is investigated, related to HOS methods, and used to identify levels of increasing nonlinear aeroelastic response. Royal Australian Air Force (RAAF) F/A-18 flight flutter test data is presented and analyzed. The data includes high-quality measurements of forced responses and LCO phenomena. Standard power spectral density (PSD) techniques and HOS methods are applied to the data and presented. The goal of this research is to develop methods that can identify the onset of nonlinear aeroelastic phenomena, such as LCO, during flutter testing.

Silva, Walter A.↗

NASA NTRS2005

The Effect of Basis Selection on Static and Random Acoustic Response Prediction Using a Nonlinear Modal Simulation

An investigation of the effect of basis selection on geometric nonlinear response prediction using a reduced-order nonlinear modal simulation is presented. The accuracy is dictated by the selection of the basis used to determine the nonlinear modal stiffness. This study considers a suite of available bases including bending modes only, bending and membrane modes, coupled bending and companion modes, and uncoupled bending and companion modes. The nonlinear modal simulation presented is broadly applicable and is demonstrated for nonlinear quasi-static and random acoustic response of flat beam and plate structures with isotropic material properties. Reduced-order analysis predictions are compared with those made using a numerical simulation in physical degrees-of-freedom to quantify the error associated with the selected modal bases. Bending and membrane responses are separately presented to help differentiate the bases.

Rizzi, Stephen A.↗

NASA NTRS2006

Nonlinear Reduced Order Random Response Analysis of Structures with Shallow Curvature

Przekop, Adam↗

NASA NTRS2011

On the Response of a Nonlinear Structure to High Kurtosis Non-Gaussian Random Loadings

This paper is a follow-on to recent work by the authors in which the response and high-cycle fatigue of a nonlinear structure subject to non-Gaussian loadings was found to vary markedly depending on the nature of the loading. There it was found that a non-Gaussian loading having a steady rate of short-duration, high-excursion peaks produced essentially the same response as would have been incurred by a Gaussian loading. In contrast, a non-Gaussian loading having the same kurtosis, but with bursts of high-excursion peaks was found to elicit a much greater response. This work is meant to answer the question of when consideration of a loading probability distribution other than Gaussian is important. The approach entailed nonlinear numerical simulation of a beam structure under Gaussian and non-Gaussian random excitations. Whether the structure responded in a Gaussian or non-Gaussian manner was determined by adherence to, or violations of, the Central Limit Theorem. Over a practical range of damping, it was found that the linear response to a non-Gaussian loading was Gaussian when the period of the system impulse response is much greater than the rate of peaks in the loading. Lower damping reduced the kurtosis, but only when the linear response was non-Gaussian. In the nonlinear regime, the response was found to be non-Gaussian for all loadings. The effect of a spring-hardening type of nonlinearity was found to limit extreme values and thereby lower the kurtosis relative to the linear response regime. In this case, lower damping gave rise to greater nonlinearity, resulting in lower kurtosis than a higher level of damping.

Rizzi, Stephen A.↗

NASA NTRS2011

Nonlinear Oscillators in Space Physics

We discuss dynamical systems that produce an oscillation without an external time dependent source. Numerical results are presented for nonlinear oscillators in the Em1h's atmosphere, foremost the quasi-biennial oscillation (QBOl. These fluid dynamical oscillators, like the solar dynamo, have in common that one of the variables in a governing equation is strongly nonlinear and that the nonlinearity, to first order, has particular form. of 3rd or odd power. It is shown that this form of nonlinearity can produce the fundamental li'equency of the internal oscillation. which has a period that is favored by the dynamical condition of the fluid. The fundamental frequency maintains the oscillation, with no energy input to the system at that particular frequency. Nonlinearities of 2nd or even power could not maintain the oscillation.

Lester,Daniel↗

NASA NTRS2011

The Nonlinear Magnetosphere: Expressions in MHD and in Kinetic Models

Like most plasma systems, the magnetosphere of the Earth is governed by nonlinear dynamic evolution equations. The impact of nonlinearities ranges from large scales, where overall dynamics features are exhibiting nonlinear behavior, to small scale, kinetic, processes, where nonlinear behavior governs, among others, energy conversion and dissipation. In this talk we present a select set of examples of such behavior, with a specific emphasis on how nonlinear effects manifest themselves in MHD and in kinetic models of magnetospheric plasma dynamics.

Hesse, Michael↗

NASA NTRS2012

Nonlinear Reduced-Order Simulation Using An Experimentally Guided Modal Basis

A procedure is developed for using nonlinear experimental response data to guide the modal basis selection in a nonlinear reduced-order simulation. The procedure entails using nonlinear acceleration response data to first identify proper orthogonal modes. Special consideration is given to cases in which some of the desired response data is unavailable. Bases consisting of linear normal modes are then selected to best represent the experimentally determined transverse proper orthogonal modes and either experimentally determined inplane proper orthogonal modes or the special case of numerically computed in-plane companions. The bases are subsequently used in nonlinear modal reduction and dynamic response simulations. The experimental data used in this work is simulated to allow some practical considerations, such as the availability of in-plane response data and non-idealized test conditions, to be explored. Comparisons of the nonlinear reduced-order simulations are made with the surrogate experimental data to demonstrate the effectiveness of the approach.

Rizzi, Stephen A.↗

NASA NTRS2013

Nonlinear Modeling by Assembling Piecewise Linear Models

To preserve nonlinearity of a full order system over a parameters range of interest, we propose a simple modeling approach by assembling a set of piecewise local solutions, including the first-order Taylor series terms expanded about some sampling states. The work by Rewienski and White inspired our use of piecewise linear local solutions. The assembly of these local approximations is accomplished by assigning nonlinear weights, through radial basis functions in this study. The efficacy of the proposed procedure is validated for a two-dimensional airfoil moving at different Mach numbers and pitching motions, under which the flow exhibits prominent nonlinear behaviors. All results confirm that our nonlinear model is accurate and stable for predicting not only aerodynamic forces but also detailed flowfields. Moreover, the model is robustness-accurate for inputs considerably different from the base trajectory in form and magnitude. This modeling preserves nonlinearity of the problems considered in a rather simple and accurate manner.

Yao, Weigang↗

NASA NTRS2016

A Nonlinear Modal Aeroelastic Solver for FUN3D

A nonlinear structural solver has been implemented internally within the NASA FUN3D computational fluid dynamics code, allowing for some new aeroelastic capabilities. Using a modal representation of the structure, a set of differential or differential-algebraic equations are derived for general thin structures with geometric nonlinearities. ODEPACK and LAPACK routines are linked with FUN3D, and the nonlinear equations are solved at each CFD time step. The existing predictor-corrector method is retained, whereby the structural solution is updated after mesh deformation. The nonlinear solver is validated using a test case for a flexible aeroshell at transonic, supersonic, and hypersonic flow conditions. Agreement with linear theory is seen for the static aeroelastic solutions at relatively low dynamic pressures, but structural nonlinearities limit deformation amplitudes at high dynamic pressures. No flutter was found at any of the tested trajectory points, though LCO may be possible in the transonic regime.

Goldman, Benjamin D.↗