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

Via new perspectives, fur the time dimension, the I,resent exposition overv iews new and recent advances describing a standardized formal theory toward s the evolution, classification, characterization and generic design of tim e 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 o f computational algorithms and their subsequent classification and characte rization. And, the overall developments are new and significantly different from the way traditional modal type and a nide variety of step-by-step tim e marching approaches which we are mostly familiar with have been developed and described in the research literature and in standard test books over t he years. The theoretical ideas and basis towards tho evolution of a genera lized 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 i s herein hypothesized that integral operators and the associated representa tions and a wide variety of the so-called integration operators pertain tu and emanate from the same family, with the burden which is being carried by a virtual field or weighted time field specifically introduced fur the tim e discretization is strictly enacted in a mathematically consistent manner so as to first permit obtaining the adjoint operator of the original semi-d iscretized equation system. Subsequently, the selection or burden carried b y the virtual or weighted time fields originally introduced tu facilitate t he time discretization process determines the formal development and outcom e of "exact integral operators", "approximate integral operators", includin g providing avenues leading to the design of new computational algorithms w hich have not been exploited and/or explored to-date and the recovery of mo st uf the existing algorithms, and also bridging the relationships systemat ically leading to the evolution of a nide variety of "integration operators " Thus, the overall developments not only serve as a prelude towards the fo rmal 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 standardized W-p-Family, and emanate from the pri nciples presented herein. The developments first leading to integral operat ors in time, and the resulting consequences then systematically leading to nut only providing new avenues but additionally also explaining a wide vari ety of generalized integration operators in time of which single-step time integration operators and various widely recognized algorithms: which we ar e familiar are simply subsets, the associated multi-step time integration o perators, and a class of finite element in time integration operators, and their relationships are particularly addressed. The theoretical design deve lopments encompass and explain a variety of time discretized operators, the recovery of various original methods of algorithmic development, and the d evelopment of new computational algorithms which have not been exploited an d/or explored to-date, and furthermore, permit time discretized operators t o 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 forma l theory of development of time discretized operators and forum fur selecti ng and identifying time discretized operators, but also permit lucid commun ication when recurring tu various time discretized operators. That which co nstitutes characterization of time discretized. operators are the so-called DNA algorithmic markers which essentially comprise of both: (i) the weight ed time fields introduced for enacting the time discretization process, and (ii) the corresponding conditions (if any) these weighted time fields impo se (dictate) upon the approximations for the dependent field variables and updates ill the theoretical development of time discretized operators. As s uch, 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.