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NASA NTRS1997

A Multi-Resolution Nonlinear Mapping Technique for Design and Analysis Application

This report describes a nonlinear mapping technique where the unknown static or dynamic system is approximated by a sum of dimensionally increasing functions (one-dimensional curves, two-dimensional surfaces, etc.). These lower dimensional functions are synthesized from a set of multi-resolution basis functions, where the resolutions specify the level of details at which the nonlinear system is approximated. The basis functions also cause the parameter estimation step to become linear. This feature is taken advantage of to derive a systematic procedure to determine and eliminate basis functions that are less significant for the particular system under identification. The number of unknown parameters that must be estimated is thus reduced and compact models obtained. The lower dimensional functions (identified curves and surfaces) permit a kind of "visualization" into the complexity of the nonlinearity itself.

Phan, Minh Q.↗

NASA NTRS1999

Nonlinear Response of Composite Panels Under Combined Acoustic Excitation and Aerodynamic Pressure

A finite element formulation is presented for the analysis of large deflection response of composite panels subjected to aerodynamic pressure- at supersonic flow and high acoustic excitation. The first-order shear deformation theory is considered for laminated composite plates, and the von Karman nonlinear strain-displacement relations are employed for the analysis of large deflection panel response. The first-order piston theory aerodynamics and the simulated Gaussian white noise are employed for the aerodynamic and acoustic loads, respectively. The nonlinear equations of motion for an arbitrarily laminated composite panel subjected to a combined aerodynamic and acoustic pressures are formulated first in structure node degrees-of-freedom. The system equations are then transformed and reduced to a set of coupled nonlinear equations in modal coordinates. Modal participation is defined and the in-vacuo modes to be retained in the analysis are based on the modal participation values. Numerical results include root mean square values of maximum deflections, deflection and strain response time histories, probability distributions, and power spectrum densities. Results showed that combined acoustic and aerodynamic loads have to be considered for panel analysis and design at high dynamic pressure values.

Abdel-Motagaly, K.↗

NASA NTRS1998

A Multi-Resolution Nonlinear Mapping Technique for Design and Analysis Applications

Phan, Minh Q.↗

NASA NTRS1999

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations That Arise from Nonlinear Two-Point Boundary Value Problems

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + definite integral of g(x, t)F(t,y(t))dt with limits between 0 and 1,0 less than or equal to x les than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integral equations arise, e.g., when one applied Green's function techniques to nonlinear two-point boundary value problems of the form y "(x) =f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and y(l) = y(sub l), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trepezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal rule, thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Sidi, Avram↗

NASA NTRS1998

Constraints on Nonlinear and Stochastic Growth Theories for Type 3 Solar Radio Bursts from the Corona to 1 AU

Existing, competing theories for coronal and interplanetary type III solar radio bursts appeal to one or more of modulational instability, electrostatic (ES) decay processes, or stochastic growth physics to preserve the electron beam, limit the levels of Langmuir-like waves driven by the beam, and produce wave spectra capable of coupling nonlinearly to generate the observed radio emission. Theoretical constraints exist on the wavenumbers and relative sizes of the wave bandwidth and nonlinear growth rate for which Langmuir waves are subject to modulational instability and the parametric and random phase versions of ES decay. A constraint also exists on whether stochastic growth theory (SGT) is appropriate. These constraints are evaluated here using the beam, plasma, and wave properties (1) observed in specific interplanetary type III sources, (2) predicted nominally for the corona, and (3) predicted at heliocentric distances greater than a few solar radii by power-law models based on interplanetary observations. It is found that the Langmuir waves driven directly by the beam have wavenumbers that are almost always too large for modulational instability but are appropriate to ES decay. Even for waves scattered to lower wavenumbers (by ES decay, for instance), the wave bandwidths are predicted to be too large and the nonlinear growth rates too small for modulational instability to occur for the specific interplanetary events studied or the great majority of Langmuir wave packets in type III sources at arbitrary heliocentric distances. Possible exceptions are for very rare, unusually intense, narrowband wave packets, predominantly close to the Sun, and for the front portion of very fast beams traveling through unusually dilute, cold solar wind plasmas. Similar arguments demonstrate that the ES decay should proceed almost always as a random phase process rather than a parametric process, with similar exceptions. These results imply that it is extremely rare for modulational instability or parametric decay to proceed in type III sources at any heliocentric distance: theories for type III bursts based on modulational instability or parametric decay are therefore not viable in general. In contrast, the constraint on SGT can be satisfied and random phase ES decay can proceed at all heliocentric distances under almost all circumstances. (The contrary circumstances involve unusually slow, broad beams moving through unusually hot regions of the Corona.) The analyses presented here strongly justify extending the existing SGT-based model for interplanetary type III bursts (which includes SGT physics, random phase ES decay, and specific electromagnetic emission mechanisms) into a general theory for type III bursts from the corona to beyond 1 AU. This extended theory enjoys strong theoretical support, explains the characteristics of specific interplanetary type III bursts very well, and can account for the detailed dynamic spectra of type III bursts from the lower corona and solar wind.

Cairns, Iver H.↗

NASA NTRS1999

Improving the Accuracy of Quadrature Method Solutions of Fredholm Integral Equations that Arise from Nonlinear Two-Point Boundary Value Problems

In this paper we are concerned with high-accuracy quadrature method solutions of nonlinear Fredholm integral equations of the form y(x) = r(x) + integral(0 to 1) g(x,t) F(t, y(t)) dt, 0 less than or equal to x less than or equal to 1, where the kernel function g(x,t) is continuous, but its partial derivatives have finite jump discontinuities across x = t. Such integrals equations arise, e.g., when one applies Green's function techniques to nonlinear two-point boundary value problems of the form U''(x) = f(x,y(x)), 0 less than or equal to x less than or equal to 1, with y(0) = y(sub 0) and g(l) = y(sub 1), or other linear boundary conditions. A quadrature method that is especially suitable and that has been employed for such equations is one based on the trapezoidal rule that has a low accuracy. By analyzing the corresponding Euler-Maclaurin expansion, we derive suitable correction terms that we add to the trapezoidal thus obtaining new numerical quadrature formulas of arbitrarily high accuracy that we also use in defining quadrature methods for the integral equations above. We prove an existence and uniqueness theorem for the quadrature method solutions, and show that their accuracy is the same as that of the underlying quadrature formula. The solution of the nonlinear systems resulting from the quadrature methods is achieved through successive approximations whose convergence is also proved. The results are demonstrated with numerical examples.

Sidi, Avram↗

NASA NTRS2000

Nonlinear Transient Problems Using Structure Compatible Heat Transfer Code

The report documents the recent effort to enhance a transient linear heat transfer code so as to solve nonlinear problems. The linear heat transfer code was originally developed by Dr. Kim Bey of NASA Largely and called the Structure-Compatible Heat Transfer (SCHT) code. The report includes four parts. The first part outlines the formulation of the heat transfer problem of concern. The second and the third parts give detailed procedures to construct the nonlinear finite element equations and the required Jacobian matrices for the nonlinear iterative method, Newton-Raphson method. The final part summarizes the results of the numerical experiments on the newly enhanced SCHT code.

Hou, Gene↗

NASA NTRS1999

Cycle 2 Nonlinear Design Optimization Analytical Cross Checks

The objectives of the Cycle 2 Nonlinear Design Optimization Anlaytical Cross Checks are to: 1) Understand the variability in the predicted performance levels of the nonlinear designs arising from the use of different inviscid (full potential/Euler) and viscous (Navier-Stokes) analysis methods; and 2) Provide the information required to allow the performance levels of all three designs to be validated using the data from the NCV (nonlinear Cruise Validation) model test.

Bencze, Dan↗

NASA NTRS1998

Investigation of Third Order Optical Nonlinearity and Reverse Saturable Absorption of Octa-alkoxy Metallophthalocyanines

In recent years, there has been a growing interest in the development of passive optical power limiters for the protection of the human eye and solid-state sensors from damage caused by energetic light pulses and also for other switching applications. One of the key issues involved is the search for appropriate materials that show effective reverse saturable absorption. Phthalocyanines seem to be good candidates for such applications because of their higher third order nonlinearity and the unique electronic absorption characteristics. A series of 1,4,8,11,15, 18,22,25-octa-alkoxy metallophthalocyanines containing various central metal atoms such as zinc, copper, palladium, cobalt and nickel were characterized for their third order nonlinearity and for their nonlinear absorptive properties to evaluate their suitability to function as reverse saturable absorbers.

Sanghadasa, Mohan↗

NASA NTRS1998

Syntheses of Octasubstituted Metal Phthalocyanines for Nonlinear Optics

Many organic materials can be used as nonlinear optical media. Phthalocyanines are of special interest because they show an unusually large third order nonlinear response, they are thermally and photochemically stable and they can be formed into oriented thin films (Langmuir-Blodgett films). They also can be easily complexed by a large variety of metals, which place them at the interface between organics and organometallics, and allows for fine tuning of the macro cycle electronic properties by the coordinated metal and substituent groups. A series of 1,4,8,11,15,18,22,25-octaalkoxy metal-free and metal phthalocyanines and 2,3,9,10,16,17,23,24-octaalkoxy metal phthalocyanines has been synthesized. Their nonlinear optical properties have been measured. The physical properties of all the phthalocyanines synthesized in this work are subject to both acid and solvent effects.

Guo, Huaisong↗

NASA NTRS1998

Fabrication and Characterization of Thin Film Ion Implanted Composite Materials for Integrated Nonlinear Optical Devices

Ion implantation has been shown to produce a high density of metal colloids within the layer regions of glasses and crystalline materials. The high-precipitate volume fraction and small size of metal nanoclusters formed leads to values for the third-order susceptibility much greater than those for metal doped solids. This has stimulated interest in use of ion implantation to make nonlinear optical materials. On the other side, LiNbO3 has proved to be a good material for optical waveguides produced by MeV ion implantation. Light confinement in these waveguides is produced by refractive index step difference between the implanted region and the bulk material. Implantation of LiNbO3 with MeV metal ions can therefore result into nonlinear optical waveguide structures with great potential in a variety of device applications. We describe linear and nonlinear optical properties of a waveguide structure in LiNbO3-based composite material produced by silver ion implantation in connection with mechanisms of its formation.

Sarkisov, S.↗

NASA NTRS2000

Nonlinear Shock Acceleration and Photon Emission in Supernova Remnants

We have extended a simple model of nonlinear diffusive shock acceleration (Berezhko & Ellison 1999: Ellison &, Berezhko 1999a) to include the injection and acceleration of electrons and the production of photons from bremsstrahlung, synchrotron, inverse Compton, and pion-decay processes. We argue that, the results of this model, which is simpler to use than more elaborate ones, offer a significant improvement, over test-particle, power-law spectra which are often used in astrophysical applications of diffusive shock acceleration. With an evolutionary supernova remnant (SNR) model to obtain shock parameters as functions of ambient interstellar medium parameters and time, we predict broad-band continuum photon emission from supernova remnants in general, and SN1006 in particular, showing that our results compare well with the more complete time-dependent and spherically symmetric nonlinear model of Berezhko, Ksenofontov, & Petukhov (1999a). We discuss the implications nonlinear shock acceleration has for X-ray line emission, and use our model to describe how ambient conditions determine the TeV/radio flux ratio, an important parameter for gamma-ray observations of radio SNRs.

Ellison, Donald C.↗

NASA NTRS1999

Piezoelectric Field Enhanced Second-Order Nonlinear Optical Susceptibilities in Wurtzite GaN/AlGaN Quantum Wells

Second-order nonlinear optical processes including second-harmonic generation, optical rectification, and difference-frequency generation associated with intersubband transitions in wurtzite GaN/AlGaN quantum well (QW) are investigated theoretically. Taking into account the strain-induced piezoelectric (PZ) effects, we solve the electronic structure of the QW from coupled effective-mass Schrodinger equation and Poisson equation including the exchange-correlation effect under the local-density approximation. We show that the large PZ field in the QW breaks the symmetry of the confinement potential profile and leads to large second-order susceptibilities. We also show that the interband optical pump-induced electron-hole plasma results in an enhancement in the maximum value of the nonlinear coefficients and a redshift of the peak position in the nonlinear optical spectrum. By use of the difference-frequency generation, THz radiation can be generated from a GaN/Al(0.75)Ga(0.25)N with a pump laser of 1.55 micron.

Liu, Ansheng↗

NASA NTRS1999

Nonlinear Effects in Single-Pass ICRF Heating

The Variable Specific Impulse Magnetoplasma Rocket (VASIMR) concept employs Ion Cyclotron Resonant Frequency (ICRF) heating as the main power deposition mechanism. Since the ions accelerate to the full energy in a single pass through the cyclotron resonance, their response to the RF-field will be essentially nonlinear - hence the motivation to amend the commonly used linear approach to the problem. In a collisionless plasma, the energy gain of an accelerated ion is limited by the time the particle spends at the resonance. This time is affected by: (1) incident flow velocity, (2) longitudinal grad B force, (3) ambipolar electric field, and (4) ponderomotive force of the RF-field. Our analysis shows that the grad B force is the dominant factor at low to moderate levels of RF-power. We present nonlinear scaling for the energy gain and the absorption efficiency with RF-power and plasma parameters. We also demonstrate that the nonlinear regime exhibits a steep decrease in the plasma density at the resonance.

Arefiev, A. V.↗

NASA NTRS1999

Weakly Nonlinear Description of Parametric Instabilities in Vibrating Flows

This project focuses on the effects of weak dissipation on vibrational flows in microgravity and in particular on (a) the generation of mean flows through viscous effects and their reaction on the flows themselves, and (b) the effects of finite group velocity and dispersion on the resulting dynamics in large domains. The basic mechanism responsible for the generation of such flows is nonlinear and was identified by Schlichting [21] and Longuet-Higgins. However, only recently has it become possible to describe such flows self-consistently in terms of amplitude equations for the parametrically excited waves coupled to a mean flow equation. The derivation of these equations is nontrivial because the limit of zero viscosity is singular. This project focuses on various aspects of this singular problem (i.e., the limit C equivalent to (nu)((g)(h(exp 3)))exp -1/2 << 1,where nu is the kinematic viscosity and h is the liquid depth) in the weakly nonlinear regime. A number of distinct cases is identified depending on the values of the Bond number, the size of the nonlinear terms, distance above threshold and the length scales of interest. The theory provides a quantitative explanation of a number of experiments on the vibration modes of liquid bridges and related experiments on parametric excitation of capillary waves in containers of both small and large aspect ratio. The following is a summary of results obtained thus far.

Knobloch, E.↗

NASA NTRS1999

Signatures of Nonlinear Waves in Coronal Plumes and Holes

In recent Ultraviolet Coronagraph Spectrometer/Solar and Heliospheric Observatory (UVCS/SOHO) White Light Channel (WLC) observations we found quasi-periodic variations in the polarized brightness (pB) in the polar coronal holes at heliocentric distances of 1.9-2.45 solar radii. The motivation for the observation is the 2.5D Magnetohydrodynamics (MHD) model of solar wind acceleration by nonlinear waves, that predicts compressive fluctuations in coronal holes. To help identify the waves observed with the UVCS/WLC we model the propagation and dissipation of slow magnetosonic waves in polar plumes using 1D MHD code in spherical geometry, We find that the slow waves nonlinearly steepen in the gravitationally stratified plumes. The nonlinear steepening of the waves leads to enhanced dissipation due to compressive viscosity at the wave-fronts.

Ofman, Leon↗

NASA NTRS2001

A Coupled Nonlinear Spacecraft Attitude Controller/Observer With an Unknown Constant Gyro Bias

A nonlinear control scheme for attitude control of a spacecraft is combined with a nonlinear gyro bias observer for the case of constant gyro bias. The closed loop system is proven to be globally stable, with zero tracking error, thus proving a separation principle for the given system. The nonlinear observer incorporates persistency of excitation, resulting in exponential convergence of the gyro bias error.

Deutschmann, Julie↗

NASA NTRS1996

Some Aspects of Nonlinear Dynamics and CFD

The application of nonlinear dynamics to improve the understanding of numerical uncertainties in computational fluid dynamics (CFD) is reviewed. Elementary examples in the use of dynamics to explain the nonlinear phenomena and spurious behavior that occur in numerics are given. The role of dynamics in the understanding of long time behavior of numerical integrations and the nonlinear stability, convergence, and reliability of using time-marching approaches for obtaining steady-state numerical solutions in CFD is explained. The study is complemented with examples of spurious behavior observed in CFD computations.

Yee, Helen C.↗