The Time Dimension and a Unified Mathematical Framework for First-Order Parabolic Systems

Tamma, Kumar K.; Zhou, Xiangmin; Kanapady, Ramdev

doi:10.1080/104077902753541005

Cited by 13 publications

(12 citation statements)

References 21 publications

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“…However, there exists a mapping relation between the time discontinuous operators to the corresponding Type 2 algorithm just as in the time continuous case. Along the same lines, it has been shown that the higher order methods such as the Runge-Kutta integration operators which also pertain to the classification of the Type 3 k q are equivalent to the integration operators emanating from the generalized weighted residual approach which are of the Type 3 k q since they both have the identical parent Type 2 k (p, q) algorithm [53,55]. The LMS methods are of the lowest order (a maximum of second-order time accuracy with unconditional stable feature) in this Type 3 k q classification with a single system single solve algorithmic structure; all the other higher-order of accuracy methods also pertain to the Type 3 k q classification and naturally involve more than a single solution step or more than a single system size as described earlier based on the relation of k and q regardless of the method pertaining to the Type 3 k q classification; however, this classification contains algorithmic structure of matrices with maximum power of only one.…”

Section: The Unified Theory Underlying Computational Algorithms For Tmentioning

confidence: 97%

“…A variety of existing algorithms in the literature pertain to the Type 3 k q classification where it is computationally easy to implement, and to-date, are mostly being perceived and/or are considered as unique and/or unrelated; however, their novelty (if any) and/or associated relationships now can be clearly understood through their identical parent Type 2 k (p, q) algorithm. For example, time discontinuous integration operators are being considered as a unique class of time operators [50][51][52] although they pertain to the classification of the Type 3 k q , however, it has been shown that these time discontinuous integration operators have equivalence to the time continuous integration operators which also pertain to the same classification of the Type 3 k q since they both have the identical parent Type 2 k (p, q) algorithm [53,54]. It is to be noted that in the context of the Type 1 and Type 2 classification, the notion of time discontinuity is not meaningful.…”

Section: The Unified Theory Underlying Computational Algorithms For Tmentioning

confidence: 98%

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Algorithms by design with illustrations to solid and structural mechanics/dynamics

Zhou

Tamma

2006

Numerical Meth Engineering

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SUMMARYA novel procedure, concepts, and new ideas to tailor and design time operators under the notion of algorithms by design is formulated in this exposition with emphasis on applications to the broad area of computational mechanics, but with focus on solid and structural mechanics/dynamics as an illustration. The algorithms by design concepts capitalize upon: (i) the recently developed unified theory underlying computational algorithms (Int. J. Numer. Meth. Engng 2004; 59:597-668), and (ii) newly established design spaces and algorithmic measures for evaluating the quality of computational algorithms (Int. J. Numer. Meth. Engng 2005; 64:1841-1870). As a step in the forward direction, in this exposition we embark upon some challenging tasks with the objective to advance, tailor, and foster the design of computational algorithms for time-dependent problems with desired and/or improved algorithmic attributes in the sense of accuracy, stability and other characteristics including algorithmic complexity in a well educated manner. The design process for computational algorithms is explained in the sense of the algorithms by design concepts via selected numerical illustrations of practical scenarios encountered in solid and structural mechanics/dynamics applications.

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Section: The Unified Theory Underlying Computational Algorithms For Tmentioning

confidence: 97%

Section: The Unified Theory Underlying Computational Algorithms For Tmentioning

confidence: 98%

Algorithms by design with illustrations to solid and structural mechanics/dynamics

Zhou

Tamma

2006

Numerical Meth Engineering

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“…For example, the LMS methods [7], the Tarnow-Simo algorithm [16], the parallel sub-stepping algorithm [17], and the class of algorithms in References [18][19][20] belong to this classification. The Runge-Kutta methods that are customarily implemented with various stages in the solution process also pertain to this design space of the Type 3 k q classification of algorithms [4,21], although, they emanate from the design space of the Type 2 classification. One of the main results in Reference [4], is the spectral equivalence relationship that has been established between the design space of the Type 2 k (p, q) classification of algorithms and the design space of the Type 3 k q classification of algorithms.…”

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

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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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“…The above Equation (3) serves as the starting point for many of the approaches in the literature for deriving time integration algorithms relevant to not only the particular class of linear multistep methods (LMS) [1][2][3][4][5], but also various classes of high-order methods [5]. However, there still exist some fundamental drawbacks that have not been addressed to-date as discussed subsequently.…”

Section: Introductionmentioning

confidence: 99%

“…However, there still exist some fundamental drawbacks that have not been addressed to-date as discussed subsequently. Several of these can be indeed circumvented and therein avenues provided for designing new computational algorithms via recent developments described in References [2][3][4][5], and, the theoretical framework in References [2][3][4][5] seeks to fundamentally explain the underlying characteristics of various existing methods that have been derived from di erent viewpoints. Historically, in the literature to-date, various approaches such as ÿnite di erence approximations for the time derivatives, variational based approaches, weighted residuals and the like have been used to deriving various types of time integration methods associated with the class of LMS methods and various other classes of high-order methods such as Runge-Kutta methods, etc.…”

Section: Introductionmentioning

confidence: 99%

A variationally consistent framework for the design of integrator and updates of generalized single step representations for structural dynamics

Kanapady¹,

Tamma²,

Zhou³

2003

Commun. Numer. Meth. Engng.

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SUMMARYA variationally consistent framework leading to the concise design of both the 'integrator' and the associated 'updates' as related to the single step representations encompassing the so-called LMS methods for structural dynamics is described. The present paper shows for the ÿrst time, a consistent treatment involving both the 'integrator' and 'updates' that are inherent in the general context of designing the time integration process. Furthermore, the framework encompasses not only all the existing time integration algorithms that are dissipative and non-dissipative within the scope of LMS methods but also contains new optimal algorithms useful for practical applications-in the sense of accuracy, stability, numerical dissipation and dispersion, and overshoot characteristics of computational algorithms for time dependent problems encountered in structural dynamics.

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The Time Dimension and a Unified Mathematical Framework for First-Order Parabolic Systems

Cited by 13 publications

References 21 publications

Algorithms by design with illustrations to solid and structural mechanics/dynamics

Algorithms by design with illustrations to solid and structural mechanics/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 variationally consistent framework for the design of integrator and updates of generalized single step representations for structural dynamics

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