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NASA NTRS1989

A space-time discretization procedure for wave propagation problems

Higher order compact algorithms are developed for the numerical simulation of wave propagation by using the concept of a discrete dispersion relation. The dispersion relation is the imprint of any linear operator in space-time. The discrete dispersion relation is derived from the continuous dispersion relation by examining the process by which locally plane waves propagate through a chosen grid. The exponential structure of the discrete dispersion relation suggests an efficient splitting of convective and diffusive terms for dissipative waves. Fourth- and eighth-order convection schemes are examined that involve only three or five spatial grid points. These algorithms are subject to the same restrictions that govern the use of dispersion relations in the constructions of asymptotic expansions to nonlinear evolution equations. A new eighth-order scheme is developed that is exact for Courant numbers of 1, 2, 3, and 4. Examples are given of a pulse and step wave with a small amount of physical diffusion.

Davis, Sanford↗

NASA NTRS1994

Dynamic modeling of orographically induced precipitation

Local orography governs the triggering of cloud formation and the enhancement of processes such as condensation and hydrometeor nucleation and growth in mountainous regions. Intense, lengthy precipitation events are typical upwind of the topographic divide, with sharply decreasing magnitude and duration on the lee side. Differences in mean annual precipitation of several hundred percent between windward slopes of orographic barriers and adjacent valleys or lee side slopes are not unusual. Because much of the streamflow in areas such as the western United States is derived from mountainous areas that are remote and often poorly instrumented, modeling of orographic precipitation has important implications for water resources management. Models of orographically induced precipitation differ by their treatment of atmospheric dynamics and by the extent to which they rely on bulk parameterization of cloud and precipitation physics. Adiabatic ascent and a direct proportionality between efficiency and orographically magnified updrafts are the most frequent assumptions in orographic precipitation modeling. Space-time discretization (i.e., resolution) is a major issue because of the high spatial variability of orographic precipitation. For a specific storm, relative errors as large as 50 to 100% are common in the forecast/hindcast of precipitation intensity and can be even larger in the case of catastrophic storms. When monthly or seasonal timescales are used to evaluate model performance, the magnitude of such errors decreases dramatically, reaching values as low as 10 to 15%. Current research is focusing on the development of data assimilation techniques to incorporate radar and satellite observations, and on the development of aggregation and disaggregation methodologies to address the implications of modeling a multiscale problem at restricted spatial and temporal resolutions.

Barros, Ana Paula↗

NASA NTRS1990

A computational procedure for the dynamics of flexible beams within multibody systems

The dynamic analysis of three dimensional elastic beams which experience large rotational and large deformational motions are examined. The beam motion is modeled using an inertial reference for the translational displacements and a body-fixed reference for the rotational quantities. Finite strain rod theories are then defined in conjunction with the beam kinematic description which accounts for the effects of stretching, bending, torsion, and transverse shear deformations. A convected coordinate representation of the Cauchy stress tensor and a conjugate strain definition is introduced to model the beam deformation. To treat the beam dynamics, a two-stage modification of the central difference algorithm is presented to integrate the translational coordinates and the angular velocity vector. The angular orientation is then obtained from the application of an implicit integration algorithm to the Euler parameter/angular velocity kinematical relation. The combined developments of the objective internal force computation with the dynamic solution procedures result in the computational preservation of total energy for undamped systems. The present methodology is also extended to model the dynamics of deployment/retrieval of the flexible members. A moving spatial grid corresponding to the configuration of a deployed rigid beam is employed as a reference for the dynamic variables. A transient integration scheme which accurately accounts for the deforming spatial grid is derived from a space-time finite element discretization of a Hamiltonian variational statement. The computational results of this general deforming finite element beam formulation are compared to reported results for a planar inverse-spaghetti problem.

Downer, Janice Diane↗

NASA NTRS1992

Dynamics of spacecraft with deploying flexible appendages

A computational formulation for the dynamic analysis of spacecraft with deploying appendages is presented. The appendage model is based on a geometrically nonlinear beam formulation which accurately accounts for large rotational and large deformation motions. A moving finite element reference grid is incorporated within the nonlinear beam formulation to model the deployment motion. Hamilton's Law is used to formulate the general equations of motion, and a transient integration solution procedure is derived from a space-time finite element discretization of the Hamiltonian variational statement. Computational results of the methodology are presented for a classical gravity gradient stabilized satellite configuration.

Downer, Janice D.↗

NASA NTRS1992

A finite element study of incompressible flows past oscillating cylinders and aerofoils

We present our numerical results for certain unsteady flows past oscillating cylinders and aerofoils. The computations are based on the stabilized space-time finite element formulation. The implicit equation systems resulting from the space-time finite element discretizations are solved using iterative solution techniques. One of the problems studied is flow past a cylinder which is forced to oscillate in the horizontal direction. In this case we observe a change from an unsymmetric mode of vortex shedding to a symmetric one. An extensive study was carried out for the case in which a cylinder is mounted on lightly damped springs and allowed to oscillate in the vertical direction. In this case the motion of the cylinder needs to be determined as part of the solution, and under certain conditions this motion changes the vortex-shedding pattern of the flow field significantly. This nonlinear fluid-structure interaction exhibits certain interesting behavior such as 'lock-in' and 'hysteresis', which are in good agreement with the laboratory experiments carried out by other researchers in the past. Preliminary results for flow past a pitching aerofoil are also presented.

Mittal, S.↗

NASA NTRS1974

The abundances of nuclei in the cosmic radiation

The relative abundances are treated as a consequence of processes in cosmic ray transport occurring during passage of the radiation through interstellar material at high velocity. Some of the subjects mentioned are nuclear fragmentation and the production of secondary nuclei, nuclear reactions, energy loss and nuclear decay, ionization, the range-energy relation and propagation variables, capture and loss of electrons, the propagation of nuclei, the transport equation, equilibrium solutions, energy-dependent path length distribution, exponential path length distributions, discrete spectra, sources, supernovae, and the origin of the abundances. The connection between the space-time features of the sources, the material traversed, and the effects of magnetic fields is established by describing the particle-field interaction as a diffusive or random-walk process.

Reames, D. V.↗

NASA NTRS1987

Irregular and diurnal variability in asynoptic measurements of stratospheric trace species

The consequences of irregular and diurnal variability on the interpretation of asynoptic measurements of stratospheric trace species are explored. In particular, the fidelity with which the continuous behavior may be recovered from discrete asynoptic measurements is examined. Dynamically created tracer variability in the form of an advected space-time random process is discussed. Diagnostics such as power spectra and time-mean fields are derived asynoptically and compared with the true behavior. Diurnal variations of a photochemically active species are considered, describing variability in terms of a solar waveform which propagates through a latitudinal envelope. Several diagnostics of the behavior are derived asynoptically and compared with the true variability.

Salby, Murry L.↗

NASA NTRS1973

Gravitation

This textbook on gravitation physics (Einstein's general relativity or geometrodynamics) is designed for a rigorous full-year course at the graduate level. The material is presented in two parallel tracks in an attempt to divide key physical ideas from more complex enrichment material to be selected at the discretion of the reader or teacher. The full book is intended to provide competence relative to the laws of physics in flat space-time, Einstein's geometric framework for physics, applications with pulsars and neutron stars, cosmology, the Schwarzschild geometry and gravitational collapse, gravitational waves, experimental tests of Einstein's theory, and mathematical concepts of differential geometry.

Misner, C. W.↗

NASA NTRS1990

Space-time modeling using environmental constraints in a mobile robot system

Grid-based models of a robot's local environment have been used by many researchers building mobile robot control systems. The attraction of grid-based models is their clear parallel between the internal model and the external world. However, the discrete nature of such representations does not match well with the continuous nature of actions and usually serves to limit the abilities of the robot. This work describes a spatial modeling system that extracts information from a grid-based representation to form a symbolic representation of the robot's local environment. The approach makes a separation between the representation provided by the sensing system and the representation used by the action system. Separation allows asynchronous operation between sensing and action in a mobile robot, as well as the generation of a more continuous representation upon which to base actions.

Slack, Marc G.↗

NASA NTRS1991

A new finite element formulation for computational fluid dynamics. IX - Fourier analysis of space-time Galerkin/least-squares algorithms

A Fourier stability and accuracy analysis of the space-time Galerkin/least-squares method as applied to a time-dependent advective-diffusive model problem is presented. Two time discretizations are studied: a constant-in-time approximation and a linear-in-time approximation. Corresponding space-time predictor multi-corrector algorithms are also derived and studied. The behavior of the space-time algorithms is compared to algorithms based on semidiscrete formulations.

Shakib, Farzin↗

NASA NTRS1994

A new flux-conserving numerical scheme for the steady, incompressible Navier-Stokes equations

This paper is concerned with the continued development of a new numerical method, the space-time solution element (STS) method, for solving conservation laws. The present work focuses on the two-dimensional, steady, incompressible Navier-Stokes equations. Using first an integral approach, and then a differential approach, the discrete flux conservation equations presented in a recent paper are rederived. Here a simpler method for determining the flux expressions at cell interfaces is given; a systematic and rigorous derivation of the conditions used to simulate the differential form of the governing conservation law(s) is provided; necessary and sufficient conditions for a discrete approximation to satisfy a conservation law in E2 are derived; and an estimate of the local truncation error is given. A specific scheme is then constructed for the solution of the thin airfoil boundary layer problem. Numerical results are presented which demonstrate the ability of the scheme to accurately resolve the developing boundary layer and wake regions using grids which are much coarser than those employed by other numerical methods. It is shown that ten cells in the cross-stream direction are sufficient to accurately resolve the developing airfoil boundary layer.

Scott, James R.↗

NASA NTRS1989

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.

Ho, Lee-Wing↗

NASA NTRS1975

Theory of type 3b solar radio bursts

During the initial space-time evolution of an electron beam injected into the corona, the strong beam-plasma interaction occurs at the head of the beam, leading to the amplification of a quasi-monochromatic large-amplitude plasma wave that stabilizes by trapping the beam particles. Oscillation of the trapped particles in the wave troughs amplifies sideband electrostatic waves. The sidebands and the main wave subsequently decay to observable transverse electromagnetic waves through the parametric decay instability. This process gives rise to the elementary striation bursts. Owing to velocity dispersion in the beam and the density gradient of the corona, the entire process may repeat at a finite number of discrete plasma levels, producing chains of elementary bursts. All the properties of the type IIIb bursts are accounted for in the context of the theory.

Smith, R. A.↗

NASA NTRS1976

Theory of type IIIb solar radio bursts

Smith, R. A.↗

NASA NTRS1994

New Developments in the Method of Space-Time Conservation Element and Solution Element-Applications to Two-Dimensional Time-Marching Problems

A new numerical discretization method for solving conservation laws is being developed. This new approach differs substantially in both concept and methodology from the well-established methods, i.e., finite difference, finite volume, finite element, and spectral methods. It is motivated by several important physical/numerical considerations and designed to avoid several key limitations of the above traditional methods. As a result of the above considerations, a set of key principles for the design of numerical schemes was put forth in a previous report. These principles were used to construct several numerical schemes that model a 1-D time-dependent convection-diffusion equation. These schemes were then extended to solve the time-dependent Euler and Navier-Stokes equations of a perfect gas. It was shown that the above schemes compared favorably with the traditional schemes in simplicity, generality, and accuracy. In this report, the 2-D versions of the above schemes, except the Navier-Stokes solver, are constructed using the same set of design principles. Their constructions are simplified greatly by the use of a nontraditional space-time mesh. Its use results in the simplest stencil possible, i.e., a tetrahedron in a 3-D space-time with a vertex at the upper time level and other three at the lower time level. Because of the similarity in their design, each of the present 2-D solvers virtually shares with its 1-D counterpart the same fundamental characteristics. Moreover, it is shown that the present Euler solver is capable of generating highly accurate solutions for a famous 2-D shock reflection problem. Specifically, both the incident and the reflected shocks can be resolved by a single data point without the presence of numerical oscillations near the discontinuity.

Chang, Sin-Chung↗

NASA NTRS1993

Planetary circulations in the presence of transient and self-induced heating

The research program focuses on large-scale circulations and their interaction with the global convective pattern. An 11-year record of global cloud imagery and contemporaneous fields of motion and temperature have been used to investigate organized convection and coherent variability of the tropical circulation operating on intraseasonal time scales. This study provides a detailed portrait of tropical variability associated with the so-called Madden-Julian Oscillation (MJO). It reveals the nature, geographical distribution, and seasonality of discrete convective signal, which is a measure of feedback between the circulation and the convective pattern. That discrete spectral behavior has been evaluated in light of natural variability of the ITCZ associated with climatological convection. A composite signature of the MJO, based on cross-covariance statistics of cloud cover, motion, and temperature, has been constructed to characterize the lifecycle of the disturbance in terms of these properties. The composite behavior has also been used to investigate the influence the MJO exerts on the zonal-mean circulation and the involvement of the MJO in transfers of momentum between the atmosphere and the solid Earth. The aforementioned observational studies have led to the production of two animations. One reveals the convective signal in band-pass filtered OLR and compares it to climatological convection. The other is a 3-dimensional visualization of the composite lifecycle of the MJO. With a clear picture of the MJO in hand, feedback between the circulation and the convective pattern can be diagnosed meaningfully in numerical simulations. This process is being explored in calculations with the linearized primitive equations on the sphere in the presence of realistic stability and shear. The numerical framework represents climatological convection as a space-time stochastic process and wave-induced convection in terms of the vertically-integrated moisture flux convergence. In these calculations, frictional convergence near the equator emerges as a key to feedback between the circulation and the convective pattern. At low latitudes, nearly geostrophic balance in the boundary layer gives way to frictional balance. This shifts the wave-induced convection into phase with the temperature anomaly and allows the attending heating to feed back positively onto the circulation. The calculations successfully reproduce the salient features of the MJO. They are being used to understand the growth and decay phases of the composite lifecycle and the conditions that favor amplification of the MJO.

Salby, Murry L.↗

NASA NTRS1991

A new numerical framework for solving conservation laws: The method of space-time conservation element and solution element

A new numerical framework for solving conservation laws is being developed. It employs: (1) a nontraditional formulation of the conservation laws in which space and time are treated on the same footing, and (2) a nontraditional use of discrete variables such as numerical marching can be carried out by using a set of relations that represents both local and global flux conservation.

Chang, Sin-Chung↗

NASA NTRS1990

Astrophysical fractals - An overview and prospects

Different astrophysical circumstances under which fractal structures have been identified so far, or are likely to be identified in the future, are reviewed. The observed fractals can be classified into 2 main groups: (1) fractal configurations in space-time, materializing as fractals defined over the time axis at a given position in space, or over the physical configuration space at a fixed instant in time; and (2) fractals in parameter spaces. The theoretical interpretation of the origin of the spatial fractal geometry of the most conspicuous 'irregular' astronomical bodies is still wanting in the context of standard continuum models. In contrast, the less conventional discrete models (cellular automata) naturally produce such spatially fractal structures.

Perdang, J.↗