On a novel design of a new unified variational framework of discontinuous/continuous time operators of high order and equivalence

Kanapady, Ramdev; Tamma, Kumar K.

doi:10.1016/s0168-874x(03)00056-8

Cited by 12 publications

(8 citation statements)

References 41 publications

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“…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: 99%

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: 99%

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

Zhou

Tamma

2006

Numerical Meth Engineering

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“…Below, we will shortly describe the second-and high-order implicit TCG methods derived in [32,35]. In the case of zero numerical dissipation, the high-order implicit TCG methods coincide with the known high-order accurate methods presented in [36][37][38] and correspond to the diagonal of the Padé approximation table. For these methods, we will derive an exact closed-form a priori global error estimator in time in Section 3.…”

Section: Second-and High-order Implicit Tcg Methodsmentioning

confidence: 98%

A new exact, closed‐form a priori global error estimator for second‐ and higher‐order time‐integration methods for linear elastodynamics

Idesman

2011

Numerical Meth Engineering

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SUMMARYIn our recent papers, we suggested a new two-stage time-integration procedure for linear elastodynamics problems and showed that for long-term integration, time-integration methods with zero numerical dissipation are very effective for all linear elastodynamics problems, including structural dynamics, wave propagation and impact problems. In this paper, we have derived a new exact, closed-form a priori global error estimator for time integration of linear elastodynamics by the trapezoidal rule and the high-order time continuous Galerkin (TCG) methods with zero numerical dissipation (these methods correspond to the diagonal of the Padé approximation table). The new a priori global error estimator allows the selection of the size (the number) of time increments for the indicated time-integration methods at the prescribed accuracy as well as the comparison of the effectiveness of the second-and high-order TCG methods at different observation times. A numerical example shows a good agreement between theoretical and numerical results.

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“…The choice of weighting function space and trial solution function space need not be the same, which leads to many different classes of algorithms (although the general convergence, stability, and error analyses associated with the time-discontinuous Galerkin approach cannot be used). Specific examples are given in Section 4.4; for a broader perspective, including a historical perspective see Bazzi and Anderheggen (1982), Hoff and Pahl (1988a,b), Kujawski and Desai (1984), Wood (1990), Sha (2000, 2001), Kanapady and Tamma (2003a), , Zhou and Tamma (2004), Fung (1999a,b), and Fung (2000).…”

Section: Weighted Residual Methods In Timementioning

confidence: 99%

“…The series of papers by Tamma and colleagues Sha 2000, 2001;Kanapady and Tamma, 2003a) aim at a general framework for the analysis and design of time integration algorithms. By starting from the most general weighted residual approach, they demonstrate that different choices for interpolations of the dynamic fields and different choices for the weighting functions can reproduce the existing one-step and LMS algorithms that have been published.…”

Section: Time Finite Element Algorithms and Other Formsmentioning

confidence: 99%

Computational Structural Dynamics

Hulbert

2004

Encyclopedia of Computational Mechanics

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This chapter presents an overview and a state‐of‐the art compilation of time integration methods for solving problems of dynamics, with a particular focus on developments that have occurred during the past 25 years. Various classifications of time integration algorithms are provided that assist the reader in understanding the many different types of numerical methods available for solving time‐dependent problems. Time integration algorithms for linear structural dynamics are reviewed thoroughly, from classical Newmark‐type schemes, to recent developments in time finite element‐based approaches. The distinction between linear and nonlinear dynamics is articulated and numerical methods developed for direct application towards solving nonlinear dynamics probems are presented. The chapter concludes with guidelines towards selecting an appropriate time integration method, along with guidelines towards choosing appropriate time step sizes.

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On a novel design of a new unified variational framework of discontinuous/continuous time operators of high order and equivalence

Cited by 12 publications

References 41 publications

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

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

A new exact, closed‐form a priori global error estimator for second‐ and higher‐order time‐integration methods for linear elastodynamics

Computational Structural Dynamics

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