SUMMARYThe primary objectives of the present exposition are to: (i) provide a generalized uniÿed mathematical framework and setting leading to the unique design of computational algorithms for structural dynamic problems encompassing the broad scope of linear multi-step (LMS) methods and within the limitation of the Dahlquist barrier theorem (Reference [3], G. Dahlquist, BIT 1963; 3:27), and also leading to new designs of numerically dissipative methods with optimal algorithmic attributes that cannot be obtained employing existing frameworks in the literature, (ii) provide a meaningful characterization of various numerical dissipative/non-dissipative time integration algorithms both new and existing in the literature based on the overshoot behavior of algorithms leading to the notion of algorithms by design, (iii) provide design guidelines on selection of algorithms for structural dynamic analysis within the scope of LMS methods. For structural dynamics problems, ÿrst the so-called linear multi-step methods (LMS) are proven to be spectrally identical to a newly developed family of generalized single step single solve (GSSSS) algorithms. The design, synthesis and analysis of the uniÿed framework of computational algorithms based on the overshooting behavior, and additional algorithmic properties such as second-order accuracy, and unconditional stability with numerical dissipative features yields three sub-classes of practical computational algorithms: (i) zero-order displacement and velocity overshoot (U0-V0) algorithms; (ii) zero-order displacement and ÿrst-order velocity overshoot (U0-V1) algorithms; and (iii) ÿrst-order displacement and zero-order velocity overshoot (U1-V0) algorithms (the remainder involving high-orders of overshooting behavior are not considered to be competitive from practical considerations). Within each sub-class of algorithms, further distinction is made between the design leading to optimal numerical dissipative and dispersive algorithms, the continuous acceleration algorithms and the discontinuous acceleration algorithms that are subsets, and correspond to the designed placement of the spurious root at the low-frequency limit or the high-frequency limit, respectively. The conclusion and design guidelines demonstrating that the U0-V1 algorithms are only suitable for given initial velocity problems, the U1-V0 algorithms are only suitable for given initial displacement problems, * Correspondence to: K. K. Tamma and the U0-V0 algorithms are ideal for either or both cases of given initial displacement and initial velocity problems are ÿnally drawn. For the ÿrst time, the design leading to optimal algorithms in the context of a generalized single step single solve framework and within the limitation of the Dahlquist barrier that maintains second-order accuracy and unconditional stability with/without numerically dissipative features is described for structural dynamics computations; thereby, providing closure to the class of LMS methods.

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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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SUMMARYA new unified theory underlying the theoretical design of linear computational algorithms in the context of time dependent first-order systems is presented. Providing for the first time new perspectives and fresh ideas, and unlike various formulations existing in the literature, the present unified theory involves the following considerations: (i) it leads to new avenues for designing new computational algorithms to foster the notion of algorithms by design and recovering existing algorithms in the literature, (ii) describes a theory for the evolution of time operators via a unified mathematical framework, and (iii) places into context and explains/contrasts future new developments including existing designs and the various relationships among the different classes of algorithms in the literature such as linear multi-step methods, sub-stepping methods, Runge-Kutta type methods, higher-order time accurate methods, etc. Subsequently, it provides design criteria and guidelines for contrasting and evaluating time dependent computational algorithms. The linear computational algorithms in the context of first-order systems are classified as distinctly pertaining to Type 1, Type 2, and Type 3 classifications of time discretized operators. Such a distinct classification, provides for the first time, new avenues for designing new computational algorithms not existing in the literature and recovering existing algorithms of arbitrary order of time accuracy including an overall assessment of their stability and other algorithmic attributes. Consequently, it enables the evaluation and provides the relationships of computational algorithms for time dependent problems via a standardized measure based on computational effort and memory usage in terms of the resulting number of equation systems and the corresponding number of system solves. A generalized stability and accuracy limitation barrier theorem underlies the generic designs of computational algorithms with arbitrary order of accuracy and establishes guidelines which cannot be circumvented. In summary, unlike the traditional approaches and classical school of thought customarily employed in the theoretical development of computational algorithms, the unified theory underlying * Correspondence to: K. K. Tamma