An integrated design of generalized single step LMS time operators for nonlinear structural dynamics

An integrated design of generalized single step LMS methods for applications to nonlinear structural dynamics is described. The design of the mathematical framework encompasses all the traditional and new and recent optimal algorithms encompassing LMS methods, and readily permits the different a-form, v-form and d-form representations in a unique mathematical setting. As such, the theoretical developments and implementation aspects are detailed for subsequent applications to nonlinear structural dynamics problems. The developments naturally inherit a consistent treatment of nonlinear internal forces under the present umbrella of predictor multi-corrector generalized single step representations with a wide variety of algorithmic choices as options to the analyst. Within the scope of Dahlquist barrier theorem for LMS methods, the results indicate that, time integration operators with zero-order displacement and zero-order velocity overshoot behavior [U0-V0] perform ideally for general non-zero initial displacement and velocity conditions. Alternatively, for given initial displacement conditions the [U1-V0] methods could be used and for given initial velocity conditions the [U0-V1] methods could be used, although the [U0-V0] methods are ideal for general situations.