Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “time dependent behavior”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,657 records · Page 92
All sourcesNASA NTRSDOE OSTI
NASA NTRS1993

An analysis of isothermal, bithermal, and thermomechanical fatigue data of Haynes 188 and B1900+Hf by energy considerations

The low-cycle fatigue behavior of Haynes 188 and B1900+Hf under isothermal, bithermal, and thermomechanical loading conditions has been analyzed on the basis of the total hysteresis energy expended per cycle. It has been observed that in the case of isothermal fatigue the total hysteresis energy correlates well with the fatigue life. In the case of bithermal 'high rate' fatigue, for a given total hysteresis energy per cycle, the fatigue life is equal to or greater than the isothermal fatigue life at the maximum bithermal temperature. This observation could be used to establish a lower bound on life for design purposes. In one case of bithermal creep-fatigue and in thermomechanical fatigue, the life is shorter than that corresponding to the isothermal life at the maximum temperature. The energy supplied, per se, may not always give a systematic correlation with the fatigue life in the cases where time-dependent creep and environmental effects are encountered. Thus, in bithermal creep-fatigue and thermomechanical fatigue, the role of creep and environment and their dependence on the energy supplied have to be properly accounted for before the energy term can be used for life prediction.

Radhakrishnan, V. M.↗
DOE OSTI2020

Visualizing Anisotropic Oxygen Diffusion in Ceria under Activated Conditions

Oxygen reactivity plays a key role in the performance of ceria-based catalysts. Aberration-corrected transmission electron microscopy and molecular dynamics simulations were used to study the oxygen atom diffusion in ceria under activated conditions. Reactive oxygen atom and its real-time diffusion were visualized. Additionally, the interplay between cerium and oxygen atoms originating from a Coulomb interaction was revealed by the out-of-plane buckling of cerium atoms associated with oxygen transport. Anisotropic oxygen atom diffusion that depends on crystal orientations was discovered, demonstrating a preferential [001] crystallographic diffusion pathway. These findings reveal prospects for applications of anisotropic orientation-relevant fluorite-structured oxides.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Singularities in nearly uniform one-dimensional condensates due to quantum diffusion

Dissipative systems often exhibit wavelength-dependent loss rates. One prominent example is Rydberg polaritons formed by electromagnetically induced transparency, which have long been a leading candidate for studying the physics of interacting photons and also hold promise as a platform for quantum information. In this system, dissipation is in the form of quantum diffusion, i.e., proportional to k 2 (k being the wavevector) and vanishing at long wavelengths as k→0. Here, we show that one-dimensional condensates subject to this type of loss are unstable to long-wavelength density fluctuations in an unusual manner: after a prolonged period in which the condensate appears to relax to a uniform state, local depleted regions quickly form and spread ballistically throughout the system. We connect this behavior to the leading-order equation for the nearly uniform condensate—a dispersive analog to the Kardar-Parisi-Zhang equation—which develops singularities in finite time. Furthermore, we show that the wavefronts of the depleted regions are described by purely dissipative solitons within a pair of hydrodynamic equations, with no counterpart in lossless condensates. We close by discussing conditions under which such singularities and the resulting solitons can be physically realized.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS1992

Homogenization models for 2-D grid structures

In the past several years, we have pursued efforts related to the development of accurate models for the dynamics of flexible structures made of composite materials. Rather than viewing periodicity and sparseness as obstacles to be overcome, we exploit them to our advantage. We consider a variational problem on a domain that has large, periodically distributed holes. Using homogenization techniques we show that the solution to this problem is in some topology 'close' to the solution of a similar problem that holds on a much simpler domain. We study the behavior of the solution of the variational problem as the holes increase in number, but decrease in size in such a way that the total amount of material remains constant. The result is an equation that is in general more complex, but with a domain that is simply connected rather than perforated. We study the limit of the solution as the amount of material goes to zero. This second limit will, in most cases, retrieve much of the simplicity that was lost in the first limit without sacrificing the simplicity of the domain. Finally, we show that these results can be applied to the case of a vibrating Love-Kirchhoff plate with Kelvin-Voigt damping. We rely heavily on earlier results of (Du), (CS) for the static, undamped Love-Kirchhoff equation. Our efforts here result in a modification of those results to include both time dependence and Kelvin-Voigt damping.

Banks, H. T.↗
NASA NTRS1993

Discussion on selected symbiotic stars

Because of its large variety of aspects, the symbiotic phenomenon is not very suitable for a statistical treatment. It is also not clear whether symbiotic stars really represent a homogeneous group of astrophysical objects or a collection of objects of different natures but showing similar phenomena. However we are especially interested in the symbiotic phenomenon, i.e., in those physical processes occurring in the atmosphere of each individual object and in their time dependence. Such a research can be performed through the detailed analysis of individual objects. This study should be done for a time long enough to cover all the different phases of their activity, in all the spectral ranges. Since the typical time scale of the symbiotic phenomena is up to several years and decades, this represents a problem since, for instance, making astronomy outside the visual region is a quite new field of research. It was a fortunate case that a few symbiotic stars (Z And, AG Dra, CH Cyg, AX Per, and PU Vul) had undergone remarkable light variations (or 'outbursts') in recent years, which could have been followed in the space ultraviolet with IUE, and simultaneously in the optical and IR with ground-based telescopes. But, in general, the time coverage of most of the symbiotic objects is too short to have a complete picture of their behavior. In this regard, one should recall Mayall's remark about the light curve of Z And: 'Z Andromedae is another variable that shows it will require several hundred years of observations before a good analysis can be made of its variations'. This pessimistic remark should be considered as a note of caution for those involved in the interpretation of the observations. We shall discuss a number of individual symbiotic stars for which the amount of observational data is large enough to draw a rather complete picture of their general behavior and to make consistent models. We shall especially illustrate the necessary steps toward an empirical model and take the discussion of the individual objects as a useful occasion to describe different techniques of diagnosis.

Viotti, Roberto↗
NASA NTRS1997

Response of High Subsonic Jet to Nonaxisymmetric Disturbances

A model of sound generated in a high subsonic (Mach 0.9) circular jet is solved numerically in cylindrical coordinates for nonaxisymmetric disturbances. The jet is excited by transient mass injection by a finite duration pulse via a modulated ring source. The nonaxisymmetric solution is computed for long times after the initial disturbance has exited the computational domain. The long time behavior of the jet is dominated by vorticity and pressure disturbances generated at the nozzle lip and growing as they convect down-stream in the jet. These disturbances generate sound as they propagate. The primary non-axisymmetric effect that we simulate is that of a flapping mode where regions of high and low pressure alternate on opposite sides of the jet. The predominant feature of this mode is the appearance of relatively large deviations of the pressure from the ambient pressure on opposite sides of the jet and the convection of these regions downstream. We illustrate flow field, near field and far field data. Important nonaxisymmetric characteristics of the near and flow field disturbances include roughly periodic pressure elevations and depressions at opposite values of the azimuthal angle psi. These correspond to pressure disturbances propagating in the axial direction. The azimuthal velocity exhibits a sinusoidal dependence on psi with similar roughly periodic disturbances. For every azimuthal angle psi, the jet radiation peaks about 30 deg. from the jet axis, however there is now a pronounced dependence of the far field radiation pattern on psi.

Bayliss, A.↗
NASA NTRSDate not supplied

Enabling in-time Prognostics with Surrogate Modeling through Physics-enhanced Dynamic Mode Decomposition Method

Computational models provide essential quantitative tools for assessing and predicting the health and performance of physical systems. However, high-fidelity models are rarely used in real-time operations or large optimization loops, due to their time-intensive nature. A common approach to improving computational efficiency of prognosis is to employ surrogate models. Such models can significantly decrease computation time for some accuracy loss. In this context, use of Dynamic Mode Decomposition (DMD) is proposed to generate surrogate models for lithium-ion (Li-ion) battery discharge. DMD has been suggested and used successfully in the area of fluid dynamics for over a decade, but it has not been applied to the PHM domain, where far-ahead prediction of nonlinear behavior is crucial to propagate faults or predict Remaining Useful Life (RUL). For Li-ion battery health management, the standard application of DMD using only the observable quantities of interest was unable to capture the nonlinear discharge of batteries exhibited in lab testing. The Koopman theory, however, provides a mechanism to tradeoff low dimensional nonlinear models with high-dimensional linear ones in a DMD framework, by augmenting nonlinear state variables into the system representation. In this way, DMD allows for configurable simulation accuracy dependent on the dimensionality of the Koopman operator. For battery health management, we augmented the observable variables with the hidden states of a higher-fidelity physics model to build the DMD surrogate. In comparison to a high-fidelity model, the surrogate improved computational efficiency with only a minimal loss of accuracy, and enabled long-term prognostics horizons. A generalized method for this was implemented in the prog models python package.

prognostics and health management↗
DOE OSTI2022

A Mountaintop View Requires Minimal Sorting: A Faster Contour Tree Algorithm

Consider a scalar field f : M → R, where M is a triangulated simplicial mesh in R d . A level set, or contour, at value v is a connected component of f –1 (v). As v is changed, these contours change topology, merge into each other, or split. Contour trees are concise representations of f that track this contour behavior. The vertices of these trees are the critical points of f, where the gradient is zero. The edges represent changes in the topology of contours. It is a fundamental data structure in data analysis and visualization, and there is significant previous work (both theoretical and practical) on algorithms for constructing contour trees. Suppose M has n vertices, N facets, and t critical points. A classic result of Carr, Snoeyink, and Axen (2000) gives an algorithm that takes O(n log n+Nα(N)) time (where α(·) is the inverse Ackermann function). A further improvement to O(t log t + N) time was given by Chiang et al. All these algorithms involve a global sort of the critical points, a significant computational bottleneck. Unfortunately, lower bounds of Ω(t log t) also exist. We present the first algorithm that can avoid the global sort and has a refined time complexity that depends on the contour tree structure. Intuitively, if the tree is short and fat, we get significant improvements in running time. For a partition of the contour tree into a set of descending paths, P, our algorithm runs in O($\Sigma$ pϵP |p| log |p| + tα(t) + N). This is at most O(t log D + N), where D is the diameter of the contour tree. Moreover, it is O(tα(t) + N) for balanced trees, a significant improvement over the previous complexity. Our algorithm requires numerous ideas: partitioning the contour tree into join and split trees, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis. There is a crucial use of a family of binomial heaps to maintain priorities, ensuring that any comparison made is between comparable nodes of the contour tree. We also prove lower bounds showing that the $\Sigma$ pϵP |p| log |p| complexity is inherent to computing contour trees.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1990

Implicit solution of three-dimensional internal turbulent flows

The scalar form of the approximate factorization method was used to develop a new code for the solution of three-dimensional internal laminar and turbulent compressible flows. The Navier-Stokes equations in their Reynolds-averaged form are iterated in time until a steady solution is reached. Evidence is given to the implicit and explicit artificial damping schemes that proved to be particularly efficient in speeding up convergence and enhancing the algorithm robustness. A conservative treatment of these terms at domain boundaries is proposed in order to avoid undesired mass and/or momentum artificial fluxes. Turbulence effects are accounted for by the zero-equation Baldwin-Lomax turbulence model and the q-omega two-equation model. For the first, an investigation on the model behavior in case of multiple boundaries is performed. The flow in a developing S-duct is then solved in the laminar regime at Reynolds number (Re) 790 and in the turbulent regime at Re=40,000 using the Baldwin-Lomax model . The Stanitz elbow is then solved using an inviscid version of the same code at M(sub inlet)=0.4. Grid dependence and convergence rate are investigated showing that for this solver the implicit damping scheme may play a critical role for convergence characteristics. The same flow at Re=2.5x10(exp 6) is solved with the Baldwin-Lomax and the q-omega models. Both approaches showed satisfactory agreement with experiments, although the q-omega model is slightly more accurate.

Michelassi, V.↗
NASA NTRS1990

The behavior of beams of relativistic nonthermal electrons under the influence of collisions and synchrotron losses

For many astrophysical situations, such as in solar flares or cosmic gamma-ray bursts, continuum gamma rays with energies up to hundreds of MeV were observed, and can be interpreted to be due to bremsstrahlung radiation by relativistic electrons. The region of acceleration for these particles is not necessarily the same as the region in which the radiation is produced, and the effects of the transport of the electrons must be included in the general problem. Hence it is necessary to solve the kinetic equation for relativistic electrons, including all the interactions and loss mechanisms relevant at such energies. The resulting kinetic equation for non-thermal electrons, including the effects of Coulomb collisions and losses due to synchrotron emission, was solved analytically in some simple limiting cases, and numerically for the general cases including constant and varying background plasma density and magnetic field. New approximate analytic solutions are presented for collision dominated cases, for small pitch angles and all energies, synchrotron dominated cases, both steady-state and time dependent, for all pitch angles and energies, and for cases when both synchrotron and collisional energy losses are important, but for relativistic electrons. These analytic solutions are compared to the full numerical results in the proper limits. These results will be useful for calculation of spectra and angular distribution of the radiation (X rays, gamma-rays, and microwaves) emitted via synchrotron or bremsstrahlung processes by the electrons. These properties and their relevance to observations will be observed in subsequent papers.

Mctiernan, James M.↗
NASA NTRS2009

Enhancing Team Performance for Long-Duration Space Missions

Success of exploration missions will depend on skilled performance by a distributed team that includes both the astronauts in space and Mission Control personnel. Coordinated and collaborative teamwork will be required to cope with challenging complex problems in a hostile environment. While thorough preflight training and procedures will equip creW'S to address technical problems that can be anticipated, preparing them to solve novel problems is much more challenging. This presentation will review components of effective team performance, challenges to effective teamwork, and strategies for ensuring effective team performance. Teamwork skills essential for successful team performance include the behaviors involved in developing shared mental models, team situation awareness, collaborative decision making, adaptive coordination behaviors, effective team communication, and team cohesion. Challenges to teamwork include both chronic and acute stressors. Chronic stressors are associated with the isolated and confined environment and include monotony, noise, temperatures, weightlessness, poor sleep and circadian disruptions. Acute stressors include high workload, time pressure, imminent danger, and specific task-related stressors. Of particular concern are social and organizational stressors that can disrupt individual resilience and effective mission performance. Effective team performance can be developed by training teamwork skills, techniques for coping with team conflict, intracrew and intercrew communication, and working in a multicultural team; leadership and teamwork skills can be fostered through outdoor survival training exercises. The presentation will conclude with an evaluation of the special requirements associated with preparing crews to function autonomously in long-duration missions.

Orasanu, Judith M.↗
DOE OSTI2020

Numerically Relevant Timescales in the MG2 Microphysics Model

Climate models rely on parameterizations of a variety of processes in the atmospheric physics, but a common concern is that the temporal resolution is too coarse to consistently resolve the behavior that individual parameterizations are designed to capture. This study examines timescales numerically derived from the Morrison-Gettelman (MG2) microphysics as implemented within the Energy Exascale Earth System Model, version 1 (E3SMv1). Numerically relevant timescales in MG2 are derived by computing the eigenspectrum of its Jacobian. These timescales are found to often be smaller than the default 5 min time step used for MG2. The fast timescales are then heuristically connected to individual microphysics processes. By substepping a few particular rain processes within MG2, the time discretization error for those processes was considerably reduced with minimal additional expense to the overall microphysics. While this improvement has a substantial effect on the target processes and on the vertical distribution of stratiform-derived rain within E3SMv1, the overall model climate is found to not be sensitive to MG2 time step. Here, we hypothesize that this is because the surface climate does not depend strongly on certain process rates, especially MG2's rain evaporation rate.

58 GEOSCIENCES↗
NASA NTRS1993

Intensity and polarization of light scattered by size distributions of randomly oriented nonspherical particles

Calculations of light scattering by small particles are important in many diverse fields of science and engineering. In many cases of practical interest, scattering particles are nonspherical and are distributed over sizes and orientations. However, accurate light scattering computations for ensembles of nonspherical particles are difficult and time-consuming, and the literature in which such calculations are reported is rather scarce. In this paper, the T-matrix approach, as extended recently to randomly oriented particles, is used to calculate rigorously light scattering by size distributions of randomly oriented axially symmetric particles. To model the variation of particle sizes in real ensembles, we use a power law distribution typical of some terrestrial aerosols. Contour plots of intensity and degree of linear polarization for polydisperse prolate and oblate spheroids of different aspect ratios and effective equivalent-sphere size parameters from 0 to 10 are calculated and compared with calculations for equivalent spheres. The angular scattering behavior of nonspherical polydispersions is found to be greatly different from that of spheres, while the scattering properties of oblate and prolate spheroids of the same aspect ratio are similar. With increasing particle size, both intensity and polarization become more shape-dependent. In general, nonspherical particles are stronger side scatterers and weaker backscatterers than equivalent spheres. With increasing aspect ratio of nonspherical particles polarization tends to be predominantly positive. Possible effects of particle nonsphericity on optical remote sensing of atmospheric aerosols are discussed.

Mishchenko, M. I.↗
NASA NTRS1998

Investigation of Anomalous Behavior in Metallic-Based Materials Under Compressive Loading

An anomalous material response has been observed under the action of applied compressive loads in fibrous SiC/Ti (both Ti-6242 and Ti-15-3 alloys) and the monolithic nickel-base alloy IN-718 in the aged condition. The observed behavior is an increase, rather than a decrease, in the instantaneous Young's modulus with increasing load. This increase is small, but can be significant in yield surface determination tests, where an equivalent offset strain on the order of 10 micron(1 x 10(exp -6) m/m) is being used. Stiffening has been quantified by calculating offset strains from the linear elastic loading line. The offset strains associated with stiffening during compressive loading are positive and of the same order as the target offset strains in yield surface determination tests. At this time we do not have a reasonable explanation for this response nor can we identify a deformation mechanism that might cause it. On the other hand, we are not convinced that it is an artifact of the experimental procedure because a number of issues have been identified and seemingly ruled out. In fact, stiffening appears to be temperature dependent, since it decreases as the temperature increases.

Gil, Christopher M.↗
NASA NTRS2006

Development of a Countermeasure to Mitigate Postflight Locomotor Dysfunction

Astronauts returning from space flight experience locomotor dysfunction following their return to Earth. Our laboratory is currently developing a gait adaptability training program that is designed to facilitate recovery of locomotor function following a return to a gravitational environment. The training program exploits the ability of the sensorimotor system to generalize from exposure to multiple adaptive challenges during training so that the gait control system essentially learns to learn and therefore can reorganize more rapidly when faced with a novel adaptive challenge. Evidence for the potential efficacy of an adaptive generalization gait training program can be obtained from numerous studies in the motor learning literature which have demonstrated that systematically varying the conditions of training enhances the ability of the performer to learn and retain a novel motor task. These variable practice training approaches have been used in applied contexts to improve motor skills required in a number of different sports. The central nervous system (CNS) can produce voluntary movement in an almost infinite number of ways. For example, locomotion can be achieved with many different combinations of joint angles, muscle activation patterns and forces. The CNS can exploit these degrees of freedom to enhance motor response adaptability during periods of adaptive flux like that encountered during a change in gravitational environment. Ultimately, the functional goal of an adaptive generalization countermeasure is not necessarily to immediately return movement patterns back to normal. Rather the training regimen should facilitate the reorganization of available sensory and motor subsystems to achieve safe and effective locomotion as soon as possible after long duration space flight. Indeed, this approach has been proposed as a basic feature underlying effective neurological rehabilitation. We have previously confirmed that subjects participating in an adaptive generalization training program using a variety of visuomotor distortions and throwing as the dependent measure can learn to enhance their ability to adapt to a novel sensorimotor environment (Roller et al., 2001). Importantly, this increased adaptability was retained even one month after completion of the training period. Adaptive generalization has been observed in a variety of other tasks requiring sensorimotor transformations including manual control tasks and reaching (Bock et al., 2001, Seidler, 2003) and obstacle avoidance during walking (Lam and Dietz, 2004). Taken together, the evidence suggests that a training regimen exposing crewmembers to variation in locomotor conditions, with repeated transitions among states, may enhance their ability to learn how to reassemble appropriate locomotor patterns upon return from microgravity. We believe exposure to this type of training will extend crewmembers locomotor behavioral repertoires, facilitating the return of functional mobility after long duration space flight. In other words, our proposed training protocol will compel subjects to develop new behavioral solutions under varying sensorimotor demands. Over time subjects will learn to create appropriate locomotor solution more rapidly enabling acquisition of mobility sooner after long-duration space flight. A gait adaptability training program can be superimposed on nominal treadmill exercise activities thus ensuring that no additional crew time is required to perform this type of training regimen and that it can be implemented with current in-flight exercise systems available on the International Space Station.

Bloomberg, J. J.↗
NASA NTRS1989

The behavior of beams of relativistic non-thermal electrons under the influence of collisions and synchrotron losses

For many astrophysical situations, such as in solar flares or cosmic gamma-ray bursts, continuum gamma rays with energies up to hundreds of MeV were observed, and can be interpreted to be due to bremsstrahlung radiation by relativistic electrons. The region of acceleration for these particles is not necessarily the same as the region in which the radiation is produced, and the effects of the transport of the electrons must be included in the general problem. Hence it is necessary to solve the kinetic equation for relativistic electrons, including all the interactions and loss mechanisms relevant at such energies. The resulting kinetic equation for non-thermal electrons, including the effects of Coulomb collisions and losses due to synchrotron emission, was solved analytically in some simple limiting cases, and numerically for the general cases including constant and varying background plasma density and magnetic field. New approximate analytic solutions are presented for collision dominated cases, for small pitch angles and all energies, synchrotron dominated cases, both steady-state and time dependent, for all pitch angles and energies, and for cases when both synchrotron and collisional energy losses are important, but for relativistic electrons. These analytic solutions are compared to the full numerical results in the proper limits. These results will be useful for calculation of spectra and angular distribution of the radiation (x rays, gamma-rays, and microwaves) emitted via synchrotron or bremsstrahlung processes by the electrons. These properties and their relevance to observations will be observed in subsequent papers.

Mctiernan, James M.↗
DOE OSTI2022

Idealized Simulations of the Tropical Climate and Variability in the Single Column Atmosphere Model (SCAM): Radiative-Convective Equilibrium

To explore the interactions among column processes in the Community Atmosphere Model (CAM), the single-column version of CAM (SCAM) is integrated for 1000 days in radiative-convective equilibrium (RCE) with tropical values of boundary conditions, spanning a parameter or configuration space of model physics versions (v5 vs. v6), vertical resolution (standard and 60 levels), sea surface temperature (SST), and some interpretation-driven experiments. The simulated time-mean climate is reasonable, near observations and RCE of a cyclic cloud-resolving model. Updraft detrainment in the deep convection scheme produces distinctive grid-scale structures in humidity and cloud, which also interact with radiative transfer processes. These grid artifacts average out in multi-column RCE results reported elsewhere, illustrating the nuts-and-bolts interpretability that SCAM adds to the hierarchy of model configurations. Multi-day oscillations of precipitation arise from descent of warm convection-capping layers starting near the tropopause, eventually reset by a burst of convective deepening. Experiments reveal how these oscillations depend critically on an internal parameter that controls the number of neutral buoyancy levels allowed for determining cloud top and computing dilute convective available potential energy in the deep convection scheme, and merely modified a little by disabling cloud-base radiation (heating of cloud base). This strong dependence of transient behavior in 1D on this parameter will be tested in the second part of this work, in which SCAM is coupled to a parameterized dynamics of two-dimensional, linearized gravity wave, and in the 3D simulations in future study.

Meteorology & Atmospheric Sciences↗
NASA NTRS1993

FD-TD modeling of 2-D dielectric waveguides for propagation and scattering of femtosecond optical solitons

Experimentalists have produced all-optical switches capable of 100-fs responses. To adequately model such switches, nonlinear effects in optical materials (both instantaneous and dispersive) must be included. In principle, the behavior of electromagnetic fields in nonlinear dielectrics can be determined by solving Maxwell's equations subject to the assumption that the electric polarization has a nonlinear relation to the electric field. However, until our previous work, the resulting nonlinear Maxwell's equations have not been solved directly. Rather, approximations have been made that result in a class of generalized nonlinear Schrodinger equations (GNLSE) that solve only for the envelope of the optical pulses. In this paper, we present first-time calculations from the vector nonlinear Maxwell's equations of femtosecond soliton propagation and scattering, including carrier waves, in two-dimensional systems of dielectric waveguides exhibiting the Kerr and Raman quantum effects. We use the finite-difference time-domain (FD-TD) method in an extension of our 1-D work. There, in a fundamental innovation, we treated the linear and nonlinear convolutions for the electric polarization as new dependent variables. By differentiating these convolutions in the time domain, we derived an equivalent system of coupled, nonlinear second-order ODE's. These equations together with Maxwell's equations form the system that is solved to determine the electromagnetic fields in inhomogeneous nonlinear dispersive media. Backstorage in time is limited to only that needed by the time-integration algorithm for the ODE's, rather than that needed to store the time-history of the kernel functions of the convolutions (1000-10,000 time steps). Thus, a 2-D nonlinear optics model from Maxwell's equations is now feasible.

Joseph, Rose↗