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

Abstract

SUMMARYA 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 ÿrst presented with subsequent emphasis on detailed formulations of a particular class relevant to the socalled time integration approaches which belong to a much broader classiÿcation 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 signiÿcantly di erent 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-ÿeld and twoÿeld forms of representations of the semi-discretized dynamic equations of motion. Therein, the developments ÿrst 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 ÿnite 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] 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 ÿelds introduced for enacting the time discretization process, and (ii) the corresponding conditions these weighted time ÿelds impose (dictate) upon the approximations (if any) for the dependent ÿeld 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 structur...

Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics

“…In particular, the generalized integration operators for the single ÿeld form of representation for the second-order accurate GSSSS algorithms are obtained from assuming the weighted time ÿeld as a degenerated scalar polynomial function as (see Reference [2] for other details),…”

“…By deÿnition, the so-called linear multi-step methods (LMS) belong to the Type 3 k 1 classiÿcation. A particular sub-class of algorithms which only involves a single equation system with no more than a single system solve at each time step pertains to the Type 3 k 1 and termed as generalized integration operators GInO was recently described in Reference [2]. Based on a summary of the overall developments of algorithms pertaining to the Type 3 k 1 classiÿcation, we have previously described a needs assessment that a desirable computational algorithm pertaining to this classiÿcation for structural dynamics must inherit the following characteristic features:…”

“…We have recently described [1][2][3][4] the general classiÿcation of time discretized operators as pertaining to three distinct Type 1, Type 2, and Type 3 classiÿcations, wherein unlike the Types 1 and 2 classiÿcations, the altogether di erent Type 3 classiÿcation is a result of the design of computational algorithms that are ideally suitable to computer implementation and enable their practical use from the viewpoint of computational considerations. The distinct classiÿcations pertain to characterization based on: (i) the size of the system representation wherein a single system size refers to the number of degrees-of-freedom resulting from the space discretization of the problem and (ii) the associated number of system solves.…”

Generalized Implicit Multi Time Step Integration for Nonlinear Dynamic Analysis

“…Running a statically analysis in each time step would be a di cult and time-consuming procedure. Newmark-method, Wilson-scheme, HHTprocedure [1], WBZ-integration [2], generalizedmethod [3], Newmark multi-time-step approach [4], third-order time step integration [5], the Newmark complex time step [6], the time weighted function procedure [7], the generalized single step integration [8], the N?rsett time integration [9], the composite time integration [10], the higher order acceleration function [11], the implicit integration based on conserving energy and momentum [12], the Green function approach [13,14], the precise integration methods [15], the IHOA [16], and the implicit integration combined with the nite-element method [17] are some of the implicit integrations.…”

Methods of Approximation in hpk Framework for ODEs in Time Resulting from Decoupling of Space and Time in IVPs

“…The energy-conserving and decaying algorithms in structural dynamics based on  -method are considered in reference [15]. In a series of papers [16][17][18][19][20][21][22], exhaustive developments and details are presented related to time integration of ODEs .…”