Algorithms by design: Part I - On the hidden point collocation within LMS methods and implications for nonlinear dynamics applications

In computational mechanics to date, even after five decades of research dealing with integration of the linear and nonlinear dynamic equations of motion, there exists for the general cases of algorithm designs still a clear lack of a fundamental understanding regarding evaluation of these equations of motion, and how, why, and what precise time levels the integrations need to take place. This has placed major limitations on commercial code developers, and a lack of confidence in general, thereby restricting the implementation of LMS methods to only a select few that happen to be correct, but without any rigorous proofs or underlying reasons explaining the ramifications. This is extremely critical for the general cases of integration of the equations of motion, in particular for the class of Linear Multi-Step (LMS) that are implicit, involve a single solve, and are second-order accurate in time with unconditional stability (this pertains to linear dynamic situations only), since these serve as the backbone and drivers for most finite element commercial and research software. In this regard, the present Part I of the three-part exposition puts the matter to rest and provides closure, while in Part II (see Ref [1]) and Part III (see Ref [2]) we describe extensions to nonlinear dynamics applications of the basic general framework for the classes of nondissipative and controllable numerical dissipative methods, respectively. Within the confines of these LMS methods, particular attention is paid to designing algorithms that are symplectic-momentum preserving, and with enhancements to include controllable dissipative features and optimal algorithm designs. Simply for illustration of the basic ideas, readily understandable numerical examples are presented that validate the overall concepts and developments.