On the measures for evaluating time discretized operators and the consequences leading to improved algorithms by design for structural dynamics

For the fist time we describe the notion of measures that can distinguish in general, the quality of computational algorithms for time dependent problems which are formulated and outlined with particular emphasis on structural dynamics applications. From the established measures for computational algorithms, the conclusion that the most effective (in the sense of accuracy and efficiency) computational algorithm should appear in certain algorithmic representation amongst comparable algorithms is drawn. With this conclusion, and also with the notion of providing improved algorithms by design, a novel computational algorithm which departs from the traditional paradigm (in the sense of LMS methods with which we are mostly familiar with) is designed into the perspective representation of comparable algorithms and is termed as the forward displacement nonlinearly explicit L-stable (FDEL) algorithm. From the established measures for comparable algorithms, the resulting design of the FDEL formulation is then compared with the commonly advocated central difference method and the Newmark average acceleration method which pertain to the class of linear multi-step (LMS) methods for assessing both linear and nonlinear dynamic cases. The conclusions that the proposed new design of the FDEL algorithm is the most effective algorithm to-date among the class of second-order accurate algorithms for general nonlinear dynamic situations is finally drawn through rigorous numerical experiments.