Investigations Encompassing the Equations of Motion and Proper and Accurate Treatment of Algorithmic Variables: Computational Structural Dynamics and Stiff Systems

Abstract

Different from the way we have been looking in the past at developments encompassing linear and nonlinear dynamics applications via time stepping methods, and significantly different from the manner in which past developments have attempted to explain and/or justify the algorithms and their design, this is the first time we provide clear and concise advances to the field. The research and subsequent developments address: 1) How to properly integrate the equations of motion and the accurate treatment of the underlying time levels at which the computations should occur, 2) How to precisely conduct acceleration computations for general nonlinear dynamic situations and the significance of failure to follow the present propositions, and 3) Proof of concept strategies and studies to enable long term nonlinear dynamics simulations for conservative systems such that the approaches foster completion of analysis to the desired time duration of interest. All these are described in the context of illustrations to the class of LMS methods as an example to simply demonstrate the basic ideas. The proposed results help achieve required order of time accuracy of computational algorithms, required order of accuracy of acceleration fields, and foster long term dynamics of stiff nonlinear dynamics applications. Numerous numerical illustrations demonstrate the proposed developments, therein provide significant advances to the field of computational structural dynamics.