The time dimension: A theory towards the evolution, classification, characterization and design of computational algorithms for transient/ dynamic applications

Summary Via new perspectives, for the time dimension, the present exposition overviews new and recent advances describing a standardized formal theory towards the evolution, classification, characterization and generic design of time discretized operators for transient/dynamic applications. Of fundamental importance in the present exposition are the developments encompassing the evolution of time discretized operators leading to the theoretical design of computational algorithms and their subsequent classification and characterization. And, the overall developments are new and significantly different from the way traditional modal type and a wide variety of step-by-step time marching approaches which we are mostly familiar with have been developed and described in the research literature and in standard text books over the years. The theoretical ideas and basis towards the evolution of a generalized methodology and formulations emanate under the umbrella and framework and are explained via a generalized time weighted philosophy encompassing the semi-discretized equations pertinent to transient/dynamic systems. It is herein hypothesized that integral operators and the associated representations and a wide variety of the so-called integration operators pertain to and emanate from the same family, with the burden which is being carried by a virtual field or weighted time field specifically introduced for the time discretization is strictly enacted in a mathematically consistent manner so as to first permit obtaining the adjoint operator of the original semi-discretized equation system. Subsequently, the selection or burden carried by the virtual or weighted time fields originally introduced to facilitate the time discretization process determines the formal development and outcome of “exact integral operators”, “approximate integral operators”, including providing avenues leading to the design of new computational algorithms which have not been exploited and/or explored to-date and the recovery of most of the existing algorithms, and also bridging the relationships systematically leading to the evolution of a wide variety of “integration operators”. Thus, the overall developments not only serve as a prelude towards the formal developments for “exact integral operators”, but also demonstrate that the resulting “approximate integral operators” and a wide variety of “new and existing integration operators and known methods” are simply subsets of the generalizations of a standardizedW _p -Family, and emanate from the principles presented herein. The developments first leading to integral operators in time, and the resulting consequences then systematically leading to not only providing new avenues but additionally also explaining a wide variety of generalized integration operators in time of which single-step time integration operators and various widely recognized algorithms which we are familiar are simply subsets, the associated multi-step time integration operators, and a class of finite element in time integration operators, and their relationships are particularly addressed. The theoretical design developments encompass and explain a variety of time discretized operators, the recovery of various original methods of algorithmic development, and the development of new computational algorithms which have not been exploited and/or explored to-date, and furthermore, permit time discretized operators to be uniquely classified and characterized by algorithmic markers. The resulting and so-called discrete numerically assigned [DNA] algorithmic markers not only serve as a prelude towards providing a standardized formal theory of development of time discretized operators and forum for selecting and identifying time discretized operators, but also permit lucid communication when referring to various time discretized operators. That which constitutes characterization of time discretized operators are the so-called DNA algorithmic markers which essentially comprise of both: (i) the weighted time fields introduced for enacting the time discretization process, and (ii) the corresponding conditions (if any) these weighted time fields impose (dictate) upon the approximations for the dependent field variables and updates in the theoretical development of time discretized operators. As such, recent advances encompassing the theoretical design and development of computational algorithms for transient/dynamic analysis of time dependent phenomenon encountered in engineering, mathematical and physical sciences are overviewed.     

A new unified theory underlying time dependent linear first‐order systems: a prelude to algorithms by design

A new unified theory underlying the theoretical design of linear computational algorithms in the context of time dependent first‐order systems is presented. Providing for the first time new perspectives and fresh ideas, and unlike various formulations existing in the literature, the present unified theory involves the following considerations: (i) it leads to new avenues for designing new computational algorithms to foster the notion of algorithms by design and recovering existing algorithms in the literature, (ii) describes a theory for the evolution of time operators via a unified mathematical framework, and (iii) places into context and explains/contrasts future new developments including existing designs and the various relationships among the different classes of algorithms in the literature such as linear multi‐step methods, sub‐stepping methods, Runge–Kutta type methods, higher‐order time accurate methods, etc. Subsequently, it provides design criteria and guidelines for contrasting and evaluating time dependent computational algorithms. The linear computational algorithms in the context of first‐order systems are classified as distinctly pertaining to Type 1, Type 2, and Type 3 classifications of time discretized operators. Such a distinct classification, provides for the first time, new avenues for designing new computational algorithms not existing in the literature and recovering existing algorithms of arbitrary order of time accuracy including an overall assessment of their stability and other algorithmic attributes. Consequently, it enables the evaluation and provides the relationships of computational algorithms for time dependent problems via a standardized measure based on computational effort and memory usage in terms of the resulting number of equation systems and the corresponding number of system solves. A generalized stability and accuracy limitation barrier theorem underlies the generic designs of computational algorithms with arbitrary order of accuracy and establishes guidelines which cannot be circumvented. In summary, unlike the traditional approaches and classical school of thought customarily employed in the theoretical development of computational algorithms, the unified theory underlying time dependent first‐order systems serves as a viable avenue to foster the notion of algorithms by design. Copyright © 2004 John Wiley & Sons, Ltd.     

Curvilinear lattice in chiral carbon nanotubes

In this paper, the energy bands of chiral-type single-walled carbon nanotubes are studied employing a new curvilinear lattice theory and its reciprocal lattice aided by the spherical triangle theory. In this theory, the notion of a distance law is introduced as a constraint to determine the electronic characteristics of carbon nanotubes. The chiral tubes are proven to be semiconducting or metallic, depending on the diameters and helical angles of the tubes. The distance law also predicts an uncertainty for the electronic characteristics of chiral tubes due to the metallic states tightly surrounding the semiconducting states on the direct lattice, and vice versa. Results predicted by the distance law agree well with theoretical calculations of the electron density of states and published measurements.     

Analogue algorithm for parallel factorization of an exponential number of large integers: II—optical implementation

Quantum Information Processing - We report a detailed analysis of the optical realization of the analogue algorithm described in the first paper of this series (Tamma in Quantum Inf Process...     

Traffic sensing and characterization in multi-channel wireless networks for cognitive networking

Traffic sensing and characterization is an important building block of cognitive networking systems, however, it is very challenging to perform traffic characterization in multi-channel multi-radio wireless networks. Due to the presence of network traffic in multiple channels, the existing count-based packet sampling methods demand continuous capture on each channel to be effective; this requires a dedicated wireless interface per channel, and hence the existing sampling methods require a very expensive infrastructure and have poor scalability. Time-based sampling methods, on the other hand, offer a cost-effective and scalable solution by reducing the amount and cost of the resources necessary to monitor and characterize the wireless spectrum.The contributions of this paper include the following: (i) a discussion of packet sampling techniques for traffic sensing in multi-channel wireless networks, (ii) a comparison of various time-based sampling strategies using the Kullback–Leibler divergence (KLD) measure, (iii) a study on the effect of the sampling parameters on the accuracy of the sampling strategies, (iv) development of sampling accuracy graphs for easing the process of best sampling scheme selection in multi-channel wireless networks, (v) the proposal of a new metric (traffic intensity) which estimates the busyness of channels by taking into consideration not only the successfully received packets but also corrupt or broken packets, (vi) implementation of time-based sampling in a prototype traffic sensor device for multi-channel traffic sensing in IEEE 802.11 b/g networks, and (vii) characterization of a campus IEEE 802.11 network environment in a spatio-temporal–spectral fashion using sampled traffic traces collected by traffic sensors.     

SPECIALLY TAILORED TRANS FINITE-ELEMENT FORMULATIONS FOR HYPERBOLIC HEAT CONDUCTION INVOLVING NON-FOURIER EFFECTS

The phenomenon of hyperbolic heat conduction in contrast to the classical (parabolic) form of Fourier heat conduction involves thermal energy transport that propagates only at finite speeds as opposed to an infinite speed of thermal energy transport. To accommodate the finite speed of thermal wave propagation, a more precise form of heat flux law is involved, thereby modifying the heat flux originally postulated in the classical theory of heat conduction. As a consequence, for hyperbolic heat conduction problems, the thermal energy propagates with very sharp discontinuities at the wave front. The primary purpose of the present paper is to provide accurate solutions to a class of one-dimensional hyperbolic heat conduction problems involving non-Fourier effects that can precisely help understand the true response and furthermore can be used effectively for representative benchmark tests and for validating alternate schemes. As a consequence, the present paper purposely describes modeling/analysis formulations via specially tailored hybrid computations for accurately modeling the sharp discontinuities of the propagating thermal wave front. Comparative numerical test models are presented for various hyperbolic heat conduction models involving non-Fourier effects to demonstrate the present formulations.     

Applicability and evaluation of an implicit self-starting unconditionally stable methodology for the dynamics of structures

The applicability and evaluation of a new self-starting, unconditionally stable, implicit methodology of computation for the dynamics of structures is described. The methodology offers different perspectives and architecture for structural dynamics compared with the tranditional (widely advocated and commonly used) time integration methods. It is based on velocity representations and architecture and uses finite elements as the principal analysis tool for structural dynamic modeling/analysis. In particular, the dynamics of beam-type flexural models are considered, and comparative results validate and support the proposed use of the self-starting methodology of computation for the dynamics of linear/nonlinear structures. The overall effectiveness and elegance strongly support its use in most existing commercial codes.