Space-time Grid Research Articles

On the possible influence of global cosmic rotations on the setting up of a space-time grid in vicinity of the earth

As Landau and Lifshitz have shown, for a rotating observer anisotropy of light velocity in vacuum, with peak-to-peak equal to doubled linear velocity of rotation, should take place. In experiments performed so far, this anisotropy due to our secular and annual rotations has not manifested itself clearly. Here the reasons of that are discussed, and some experimental refinements are proposed. The corresponding measurable effects are comparable with the accuracy achieved in techniques setting up a space-time global grid in vicinity of the Earth.

Numerical approach to solve the time-dependent Dirac equation

We discuss how split-operator techniques may be applied to calculate normalized wave-function solutions to the three-dimensional time-dependent Dirac equation on a space-time lattice grid. This approach can be used to investigate various field-free dynamics of electron wave packets as well as the interactions of atomic electrons with space-time varying electromagnetic radiation. We demonstrate this technique by analyzing the time-resolved Klein paradox and the Zitterbewegung.

On the Spectral Ranges that Are Resolved by a Single Satellite in Exact-Repeat Sampling Mode

The sampling problem for satellite data in exact-repeat orbit configuration is treated in this paper. Specifically, for exact-repeat satellite sampling, the author seeks to solve the problem equivalent to finding the Nyquist frequency and wavenumbers for a textbook regular grid that frame a resolved spectral range within which all properly separated (assuming the data coverage is not infinite) spectral components can be distinguished (i.e., resolved) from each other, while an inside component is still indistinguishable from an infinite number of spectral components outside the range (i.e., aliasing). It is shown that there are multitudes of spectral ranges that are resolved with various degrees of uncertainty by the data: the suitable choice depends on the phenomena one wishes to observe and the noises one endeavors to avoid. The problem is idealized for applications to regions of limited latitudinal extent, so that straight lines represent satellite ground tracks well. Let X and Y be the east–west and north–south separations of parallel tracks, let T be the repeat period, and let k, l, and ω be the nonangular wavenumbers and frequency with k in the east–west direction. The spectral range that is perfectly resolved (as if the data were placed on a regular space–time grid) covers (−1/X, 1/X) in k, (−1/Y, 1/Y) in l, and [0, 1/2T) in ω. There are other spectral ranges that extend either the spatial or temporal resolution with increased uncertainty beyond the above-mentioned perfectly resolved range. The idealized problem is solved in stages, progressing from the 1D, to the 2D, to the fully 3D problem. The process is aided by a discovery that has implications that go beyond the scope of this paper. It is found that from a multidimensional regular grid one is free to introduce misalignments to all dimensions except one without incurring any penalty in spectral resolution. That is, the misaligned grid is equivalent to a perfectly aligned grid in spectral resolution. (However, the misalignment does induce far more complicated aliasing.) Thus, nonsimultaneous observations are equivalent to simultaneous ones, hence reducing the 3D problem to the 2D one. One 2D grid (the crossover grid) is equivalent to not just one but two regular 2D grids. These results are all verified numerically. An analytic proof is provided for the equivalency theorem.

On the convergence on nonrectangular grids

The convergence properties of a full discrete approximations to the convection-diffusion equation is the subject of this paper. The full discrete scheme considered is of Lagrangian type: Euler Implicit on time and centered finite difference on space, and is defined using nonrectangular grids. We analyse this scheme under smoothness conditions on nonrectangular space-time grid. The main result establish the convergence of the approximations and we prove that the assumptions on the discrete spatial nodes movement are achieved if we consider the equidistribution principle.

97/03662 Pipeline flow models of coal-water slurries

The study of coal water slurries fluidodynamics involves use of non-newtonian rheological models. In this work, the slurry flow in pipeline is described using an Herschel-Bulkley model with structure parameter. This parameter defines the thixotropic behavior of these mixtures, and this evolution is described by a kinetic equation. Two computer programs have been developed; one of these simulates flow in stationary conditions; the other one apply to an important transient phase of pipe flow: the start-up phase. The start-up has been analyzed by following a theoretical approach proposed by Cheng and Whittaker. They make a simplifying hypothesis about transition region between {open_quotes}fresh{close_quotes} fluid and fluid at rest into the pipe; with this assumption, it is possible to integrate the evolutionary equation of structure parameter, using a convenient space-time grid. Time variation of central plug size is examined - the central plug is the flow domain without shear rate - as well as the flow rate and pressure gradient evolutions. The description of calculation models is presented; further a comparison with experimental results is carried out.

Proximity in Production Networks

In a context of decreasing transport costs, quasiubiquity of quality infrastructure and the increasing importance of non-material flows, transport infra-structure appears to be less important as a location factor for industry. However, the globalization of markets and the restructuring of production systems lead to an increased role for transportation in industrial strategies. The traditional framework dealing with transportation mainly as a cost thus becomes less relevant for understanding the spatial dynamics of industry. The framework developed in this article divides production into transformation and circulation activities. Interactions between the transformation process and its environment of resources, suppliers, customers and other producers form a system of circulation of goods, information and knowledge. Transportation thus becomes a particular set of techniques of interaction in the space-time grid. The spatial dimension is introduced, not through the idea of mere spatial propinquity as a determinant of production networks, but through the concept of organizational proximity and its spatial and circulatory dimensions. The circulatory dimension of proximity describes rapid, reliable and well adapted circulation of goods and information as well as the efficient mobilization of external resources, especially of non-traded, specific resources. This dimension of proximity thus includes more than geographical accessibility. Its determinants are as much the infrastructure as a generic resource for circulation as the organization and the degree of control of flows of goods, information and people.

Detection of overall space-time clustering in a non-uniformly distributed population. DiMe Study Group.

We developed a test statistic based on an approach of Whittemore et al. (1987) to detect space-time clustering for non-infectious diseases. We extended the spatial test of Whittemore et al. by deriving conditional probabilities for Poisson distributed random variables. To combine spatial and time distances we defined a distance matrix D, where dij is the distance between the ith and jth cell in a three-dimensional space-time grid. Spatial and temporal components are controlled by a weight. By altering the weight, both marginal tests and the intermediate test can be reached. Allowing a continuum from a pure spatial to a pure temporal test, the best result will be gained by trying different weights, because the occurrence of a disease might show some temporal and some spatial tendency to cluster. We examined the behaviour of the test statistic by simulating different distributions for cases and the population. The test was applied to the incidence data of insulin-dependent diabetes mellitus in Finland. This test could be used in the analysis of data which are localized according to map co-ordinates, by addresses or postcodes. This information is important when using the Geographical Information System (GIS) technology to compute the pairwise distances needed for the proposed test.

Microearthquake analysis for hydraulic fracturing process

The hydraulic fracture technique is widely applied in the enhancement of petroleum and natural gas productions and in the development of geothermal energy. This technique is also used to create an underground fracture zone system for disposal of solid and liquid wastes. This is the most recent development in the application of industrial techniques to environmental protection scientific problems. Knowledge of mechanical properties and geometrical parameters of a hydraulic fracture zone is important for both energy resource development and safe disposal of waste. Hydraulic fracturing often induces many microearthquakes. Analysis of the spatial-temporal distribution of the induced seismicity yields the geometry of a hydraulic fracture zone, and kinetic and dynamic parameters associated with the fracture growth process. Applying a waveform correlation analysis and a space-time grid search method, we precisely determined hypocentral locations for 157 microearthquakes induced by hydraulic fracturing. Spatial distribution of the induced seismicity clearly shows the dimension and orientation of a hydraulic fracture zone. Variation of the seismicity distribution in time and space was used to infer the growth rate and direction of the fracture zone. An empirical Green’s function (EGF) method was applied to earthquake doublets to retrieve relative source time functions (RSTF’s) and to estimate source parameters, such as seismic moment, source radius, and stress drop, for larger microearthquakes. Azimuthal variation of the RSTF of a master event indicates that the source ruptured to the northwest, which aggrees with the fracture zone growth direction. Large variation of stress drops for these induced earthquakes reflects the significant heterogeneity of mechanical properties in the hydraulic fracture zone.

Direct Finite Element Computation of Periodic Adsorption Processes

This article presents a direct method for computing time-periodic solutions of adsorption processes as an alternative to prolonged dynamic simulation of the natural evolution to periodicity. Direct computation of periodicity is established by discretization on a two-dimensional space-time grid that is periodic in time. Petrov-Galerkin (SUPG) finite element approximation is applied for consistent stabilization of convective terms in the governing hyperbolic equations. Newton iteration with Gaussian elimination (frontal method) is used to solve the resulting set of nonlinear algebraic equations. Computations match exact solutions on simple adsorption cycles, and capture shock layers with as few as two elements. In its present form, the direct method is more efficient than dynamic simulation when the natural evolution to periodicity extends over hundreds of cycles, and will likely be even faster with superior discretization and solution techniques.

Generalized inversion of a global numerical weather prediction model

We construct the generalized inverse of a global numerical weather prediction (NWP) model, in order to prepare initial conditions for the model at time “t=0 hrs”. The inverse finds a weighted, least-squares best-fit to the dynamics for −24<t<0, to the previous initial condition att=−24, and to data att=−24,t=−18,t=−12 andt=0. That is, the inverse is a weak-constraint, four-dimensional variational assimilation scheme. The best-fit is found by solving the nonlinear Euler-Lagrange (EL) equations which determine the local extrema of a penalty functional. The latter is quadratic in the dynamical, initial and data residuals. The EL equations are solved using iterated representer expansions. The technique yields optimal conditioning of the very large minimization problem, which has ∼109 hydrodynamical and thermodynamical variables defined on a 4-dimensional, space-time grid.

Tracking Loop Current eddies with satellite altimetry

Geosat altimeter derived sea surface height (SSH) anomaly fields have been optimally interpolated onto a regular space time grid using both crossover data from the nonrepeating Geodetic Mission (Geosat-GM) and collinear data from the Exact Repeat Mission (Geosat-ERM). Over four years of data were collected from the combined missions, spanning the time period from April 1985 through August 1989, during which six major and at least two minor Loop Current eddies were directly observed. Eddy paths determined by automated tracking of the local maximum values in the SSH anomaly fields were compared with eddy centers estimated from drifting buoy trajectories, validating the data processing and tracking techniques. Accurate tracking of eddy centers allowed transits of 90°W to be used as a benchmark for determination of eddy shedding periods. For this data set the average period between major eddy transits was 9.8 months, with individual separation periods ranging from 6 to 14 months. The two minor eddies observed were associated with the deepest penetrations of the Loop Current into the gulf, and were nearly coincident with the shedding of the strongest major Loop Current eddies.

Digital simulation of steady state and non-steady state voltammetric responses for electrochemical reactions occurring at an inlaid microdisk electrode: Application to EC irr, EC' and CE first-order reactions

A general algorithm based on the Hopscotch method and a conformal map is used to treat electrochemical reactions with coupled homogeneous chemical reactions occurring both at conventionally sized electrodes and at ultramicroelectrodes. A change of the dimensionless parameter p = a(nFv/RTD) 1 2 , where a is the radius of the electrode, v the potential sweep rate and D the diffusion coefficient, allows us to describe properly the planar and the convergent mass transfer to an inlaid disk-shaped electrode. The dimensionless parameter λ′ = ka 2/ D, which is a function of the sum of the chemical rate constants ( k) and of the electrode radius, describes the kinetics and may be used instead of the conventional parameter λ = k(RT/nFv). A suitable choice of the space-time grid gives accurate results, even for kinetic problems affected by stiffness difficulties. The adequacy of the implementation is established by comparison with known solutions relative to the EC irr' EC' and CE mechanisms for two limiting cases, i.e. planar and radial mass transport conditions.

Convergence properties of the Runge-Kutta-Chebyshev method

The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length β proportional tos 2. The method can be applied withs arbitrarily large, which is an attractive feature due to the proportionality of β withs 2. The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for anys and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results.

A local mesh refinement algorithm for the time domain-finite difference method using Maxwell's curl equations

An efficient local mesh refinement algorithm, subdividing a computational domain to resolve fine dimensions in a time-domain-finite-difference (TD-FD) space-time grid structure, is discussed. At a discontinuous coarse-fine mesh interface, the boundary conditions for the tangential and normal field components are enforced for a smooth transition of highly nonuniform held quantities. >

Fourier Analysis of the SOR Iteration

The SOR iteration for solving linear systems of equations depends upon an overrelaxation factor omega. It is shown that for the standard model problem of Poisson's equation on a rectangle, the optimal omega and corresponding convergence rate can be rigorously obtained by Fourier analysis. The trick is to tilt the space-time grid so that the SOR stencil becomes symmetrical. The tilted grid also gives insight into the relation between convergence rates of several variants.

Converging uniaxial strain shock waves in nonlinearly locking solids under loading–unloading blasts. II: All-events method of integration along with the existing waves

A computational method is developed for the wave characteristics formulation governing transient nonlinear deformations of locking solids where waves of a moderate unloading process overtake a shocked precursor wave of large uniaxial strain. The unique feature of the proposed all-event integration scheme is that in order to evaluate the unknowns at a space–time point, not only past events are considered, but also present events surrounding the point in question are accounted for. Moreover, the solution by the proposed method propagates along all existing waves inside a variable space–time grid formed by the bicharacteristic curves associated with the chord-shock speed of the leading wave (precursor). Based on these features, the method is essentially analytical where in its final stage of development a straightforward numerical integration is required which is free of iteration procedures, stability criteria, and of any other numerical analysis.

A critique of relativity and localization

A new philosophical model makes particles and information at single points derivative. Space-time grids are not events but only ideal comparisons made by observers. Therefore the identity of space-time points and also of single particles is inherently a speculative assumption. The conservation of units can be derived and is not a foundation for events. An interaction is an actual change and can determine changed particles and a changed space-time grid from itself, both forward and backward in time. In contrast, relativity theory still retains the classical unit model in which information is localized at single points, merely positing more than one observer. One implication of the new model is that quantomechanical solutions meed not be limited by the requirements of relativity as is currently done. The model correctly predicts where difficulties should be found, and relates and explains many puzzles which are otherwise separated and inexplicable.

Propagation of weak waves in elastic-plastic and elastic-viscoplastic solids with interfaces

A numerical technique based on the method of singular surfaces has been developed for the computation of wave propagation in solids exhibiting rate-independent elastic-plastic or rate-dependent elastic-viscoplastic behavior. The von Mises yield condition and associated flow rule is taken to represent the rate-independent behavior, while the Perzyna dynamic overstress model is taken to represent the rate-dependent behavior. For 1100-0 Al, a good empirical fit with published experimental data was found to be: J21/2 − κ(Wp) = (τ0/γ0)(Wp0/J21/2) where:J2 is the second invariant of the stress deviator;k(Wp) is the static hardening curve;Wp is the plastic work and the parameter (τ0/γ0) = 0 (rate-independent model) or (80)−1 to (70)−1 MPa · s. In the numerical technique, the “connection equations” which provide relations between discontinuities in space and time derivatives lend themselves naturally to finite difference representations. A five-point space-time grid (center point coincident with the instantaneous location of the singular surface) is sufficient for the differenced form of the connection equations and suggests a natural marching scheme for the calculation of all necessary variables at each time step. Supplementing these equations which hold in the interior of the specimen are interface equations which assure continuity in stress and velocity across boundaries which separate materials with dissimilar properties. Application of the technique is made to wave propagation in pure shear for the purpose of comparing numerical predictions with relevant experimental data. The measurements of Duffyet al.[10] which are obtained from the torsional Kolsky apparatus (one dimensional torsional shear wave propagation in a thin-walled tube) were compared with predictions obtained numerically. By using the experimental input pulse history and the constitutive equation reported above, excellent agreement between the predicted and observed histories of reflected and transmitted pulses was obtained when the viscoplastic model was used. Poorer agreement was observed when the rate-independent model (τ0/γ0=0) was used. It is concluded that the Perzyna model gives good results for the behavior of 1100-0 Al at high rates of strain.

A PROPOSED STAGGERED-GRID SYSTEM FOR NUMERICAL INTEGRATION OF DYNAMIC EQUATIONS

Abstract A system is proposed for grid allocation and differencing of apparently general applicability to purely marching-type systems of equations of fluid dynamics. The method is based on casting of the equations into the conservation form, which then permits use of a staggered space-time grid system with interpolations required only in certain linear terms. The method is illustrated by application to two systems of equations, on one of which numerical experiments have been successfully performed. Advantages and drawbacks of the method are described in comparison to other currently used grid systems, and the possibility and desirability of parametric simulation of turbulent eddy exchange processes are discussed.