Time discretized operators. Part 1: towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics

Tamma, Kumar K.; Sha, Dingguo; Zhou, Xiangmin

doi:10.1016/s0045-7825(02)00515-7

Cited by 51 publications

(33 citation statements)

References 28 publications

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“…Alternatively, based on the geometrical integrator of additive decomposition of the Lie algebra approach, a new family of algorithms were also more recently designed in Reference [15] which are completely different to the single step algorithmic structure of the LMS methods in terms of the GSSSS or GS 4 methods [7,38]. And, this approach was shown to eventually lead to the design of a non-linearly explicit unconditionally stable Type 2 2 (0, 2) formulation with SSSS or S 4 algorithmic structure; however, involving a matrix power of two [9] and termed as the non-linearly explicit L-stable (EL) algorithm [10].…”

Section: Lms Methods Versus An Improved Algorithm By Design: the Linementioning

confidence: 99%

“…In contrast, in relation to the Type 2 classification of algorithms which mimic, yet further attempt to reduce the computational complexity associated with the evolution of the fundamental solution pertaining to the Type 1 classification, the subset pertaining to the Type 2 k (p, q) classification of algorithms are also in the SSSS or S 4 structure but involving matrix powers which approximate the exact amplification matrix design space, where k = max{p, q}, p and q are the maximum powers of the matrix associated with the known variables and unknown variables, respectively. For example, the algorithms described in References [9,10,15] belong to the design space of the Type 2 k (p, q) classification of algorithms. On the other hand, from the viewpoint of further enabling computational simplicity of the Type 2 classification leading to the design space of the Type 3 classification, the subset, namely, the Type 3 k q classification of algorithms are in general, a multiple-system-singlesolve (MSSS) or a single-system-multiple-solve (SSMS) algorithmic structure involving only amplification matrices with maximum power of one, where k is the number of sets of unknowns and q is the maximum number of either the number of equation systems or the number of solves.…”

Section: Introductionmentioning

confidence: 99%

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Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design—illustration to structural dynamics

Zhou

Tamma

Sha

2005

Int. J. Numer. Meth. Engng

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SUMMARYFor the first time, for time discretized operators, we describe and articulate the importance and notion of design spaces and algorithmic measures that not only can provide new avenues for improved algorithms by design, but also can distinguish in general, the quality of computational algorithms for time-dependent problems; the particular emphasis is on structural dynamics applications for the purpose of illustration and demonstration of the basic concepts (the underlying concepts can be extended to other disciplines as well). For further developments in time discretized operators and/or for evaluating existing methods, from the established measures for computational algorithms, the conclusion that the most effective (in the sense of convergence, namely, the stability and accuracy, and complexity, namely, the algorithmic formulation and algorithmic structure) computational algorithm should appear in a certain algorithmic structure of the design space amongst comparable algorithms is drawn. With this conclusion, and also with the notion of providing new avenues leading to improved algorithms by design, as an illustration, a novel computational algorithm which departs from the traditional paradigm (in the sense of LMS methods with which we are mostly familiar with and widely used in commercial software) is particularly designed into the perspective design space representation of comparable algorithms, and is termed here as the forward displacement non-linearly explicit L-stable (FDEL) algorithm which is unconditionally consistent and does not require non-linear iterations within each time step. From the established measures for comparable algorithms, simply for illustration purposes, the resulting design of the FDEL formulation is then compared with the commonly advocated explicit * Correspondence to: K. K. Tamma central difference method and the implicit Newmark average acceleration method (alternately, the same conclusion holds true against controllable numerically dissipative algorithms) which pertain to the class of linear multi-step (LMS) methods for assessing both linear and non-linear dynamic cases. The conclusions that the proposed new design of the FDEL algorithm which is a direct consequence of the present notion of design spaces and measures, is the most effective algorithm to-date to our knowledge in comparison to the class of second-order accurate algorithms pertaining to LMS methods for routine and general non-linear dynamic situations is finally drawn through rigorous numerical experiments.

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Section: Lms Methods Versus An Improved Algorithm By Design: the Linementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design—illustration to structural dynamics

Zhou

Tamma

Sha

2005

Int. J. Numer. Meth. Engng

Self Cite

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“…These lead to a new generation of a generalized family of time discretized operators with excellent algorithmic attributes which have not been adequately addressed to-date, explored or exploited and are only being recently explored. For time discretized operators of Type 2, the approximations for the dependent ÿeld variables in a given time interval are again not pertinent (see References [23][24][25][26]) based on analogous issues identiÿed previously. Both Type 1 and Type 2 classiÿcations do not pertain to LMS methods, unlike the Type 3 classiÿcation to follow.…”

Section: Classiÿcation Of Time Discretized Operatorsmentioning

confidence: 99%

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

Tamma

Zhou

Sha

2001

Numerical Meth Engineering

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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...

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“…The step-by-step implementation for an explicit method [1][2][3][4][5][6][7][8] is much simpler than that for an implicit method [1,[9][10][11][12][13], and an explicit method involves less computational efforts when compared with an implicit method. This is mainly because of the fact that an explicit method has no need to involve any iterative procedure at each time step.…”

Section: Introductionmentioning

confidence: 99%

An explicit method with improved stability property

Chang

2008

Numerical Meth Engineering

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SUMMARYA new explicit method is presented. This method has an enhanced stability property when compared with the previously published explicit method (J. Eng. Mech. 2007; 133(7):748-760), which is unconditionally stable for linear elastic and any instantaneous stiffness softening systems based on linearized analysis whereas it is only conditionally stable for any instantaneous stiffness hardening system. In contrast to the previously published explicit method, the most important improvement of the new explicit method is that it can have unconditional stability for general instantaneous stiffness hardening systems in addition to linear elastic and instantaneous stiffness softening systems. Since these stability results are obtained from a linearized stability analysis they are applicable to the non-linear systems that satisfied the simplifications used for the analysis. This explicit method is also shown to be second-order accurate. It is computationally efficient in the solution of a general structural dynamic problem, where the total response is dominated by low-frequency modes, when compared with explicit methods, such as the Newmark explicit method and the family of explicit methods developed by Tamma et al. (Comput. Methods Appl. Mech. Eng. 2003; 192:257-290), and implicit methods, such as the constant average acceleration method. Some numerical examples are used to confirm the enhanced stability property and the efficiency in computing.

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Time discretized operators. Part 1: towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics

Cited by 51 publications

References 28 publications

Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design—illustration to structural dynamics

Design spaces, measures and metrics for evaluating quality of time operators and consequences leading to improved algorithms by design—illustration to structural dynamics

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

An explicit method with improved stability property

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