Algorithms by Design: Part II—A Novel Normalized Time Weighted Residual Methodology and Design of a Family of Symplectic-Momentum Conserving Algorithms for Nonlinear Structural Dynamics

Abstract

Previously, in Part I (see [1]) of this three-part exposition, we described for the first time for the particular class of LMS methods involving a single system and a single solve within each time step, which are the predominant methods in commercial and research software, how and why the equation of motion needs to be evaluated at precise time levels (which, in general, are not known to date) for ensuring correct algorithm designs. Here in Part II of this exposition, focusing attention on this class of LMS methods, a novel displacement based normalized time weighted residual approach is presented that naturally leads to new designs of a family of numerically non-dissipative (which are of focus here) symplectic-momentum conserving time integration operators without enforcing added or artificial constraints. The objectives are to foster the simulation of nonlinear structural dynamic problems for long-term durations. For illustration, the focus is on the Saint Venant Kirchoff material although the method can be readily extended to general material models (Part III ([2]) further shows extensions of this paper to additionally include controllable numerical dissipation for nonlinear dynamics applications to address issues associated with undesirable participation of the unresolved high frequency modes induced due to spatial discretization). The theoretical developments are described for nonlinear dynamics applications via extensions of the well-known Generalized Single Step Single Solve (GSSSS) linear dynamic framework, which encompasses the class of LMS methods in the single field form. In principle, this framework comprises two distinct classifications, namely, constrained U and V algorithmic architectures (characterized by the overshoot behavior) and was originally developed for designing linear dynamic algorithms using a generalization of the classical time weighted residual method. However, the problem with the classical approach is that it fails to adequately provide proper extensions to nonlinear dynamics applications as illustrated in this paper. In contrast to the classical approach, and also different from other approaches of designing algorithms, we show in this paper, via the new displacement based normalized time weighted residual approach, how to design symplectic-momentum time integrators that are implicit, and maintain second-order time accuracy for the class of general nonlinear inertial dynamics applications. For linear dynamic framework, the present concepts readily revert to the classical approach. Numerical illustrations are presented that demonstrate the computational and implementation aspects for nonlinear structural dynamics applications.