A theory of development and design of generalized integration operators for computational structural dynamics

K. K. Tamma, X. Zhou, D. Sha

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A standardized formal theory of development/evolution, characterization and design of a wide variety of computational algorithms emanating from a generalized time weighted residual philosophy for dynamic analysis is first presented with subsequent emphasis on detailed formulations of a particular class relevant to the so-called time integration approaches which belong to a much broader classification relevant to time discretized operators. Of fundamental importance in the present exposition is the evolution of the theoretical design and the subsequent characterization encompassing a wide variety of time discretized operators, and the proposed developments are new and significantly different from the way traditional modal type and a wide variety of step-by-step time integration approaches with which we are mostly familiar 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 single-field and two-field forms of representations of the semi-discretized dynamic equations of motion. Therein, the developments first leading to integral operators in time, and the resulting consequences then systematically leading to and explaining a wide variety of generalized time integration operators of which the family of single-step time integration operators and various widely recognized and new algorithms are 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 generalized formulations not only encompass and explain a wide variety of time discretized operators and the recovery of various original methods of algorithmic development, but furthermore, naturally inherit features for providing new avenues which have not been explored and/or exploited to-date and permit time discretized operators to be uniquely characterized by algorithmic markers. The resulting and so-called discrete numerically assigned [DNA] 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 these weighted time fields impose (dictate) upon the approximations (if any) for the dependent field variables in the theoretical development of time integrators and the associated updates of the time discretized operators. Furthermore, a single analysis code which permits a variety of choices to the analyst is now feasible for performing structural dynamics computations on modern computing platforms.

Original languageEnglish (US)
Pages (from-to)1619-1664
Number of pages46
JournalInternational Journal for Numerical Methods in Engineering
Issue number7
StatePublished - 2001


  • Structural dynamics
  • Time discretized operators
  • Time integration

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