Algorithms by design: A novel normalized time weighted residual approach and design of a family of energy-momentum conserving algorithms for nonlinear structural dynamics

A novel approach termed as the hybrid displacement-strain based normalized time weighted residual approach is presented in this exposition to accurately solve nonlinear structural dynamic problems. The current work concentrates on the design of a family of conserving time operators which preserve energy, linear and angular momentum conservation properties via this new innovative approach to solve nonlinear dynamic problems. The focus of this work is on Generalized Single Step Single Solve (GSSSS) computational algorithms which encompass the entire framework of those termed as linear multi-step (LMS) methods with unconditional stability and second-order accuracy which are most commonly used for such applications. In general, there exist three roots for LMS based methods, namely, the two principal roots ρ_1∞, ρ_2∞, and the spurious root ρ_3∞. The LMS framework is comprised of two distinct algorithmic structures termed as constrained U and V algorithms. Specifically, it is shown that in the sense of LMS methods, the implementation of V0 algorithms (zero oder velocity overshoot behavior) via the hybrid displacement-strain based normalized weighted residual approach (no external constraints on energy are imposed) with ({ρ_1∞, ρ_2∞, ρ_3∞} = {1.00,1.00, any ρ_3∞ ∈ [0.00,1.00]} readily conserve energy, linear momentum and angular momentum. Although the displacement and velocity are independent of ρ_3∞, it is shown and proved that different choices of ρ_3∞ gives different values of acceleration at every time step, thus yielding a different algorithm within the family. Nevertheless, in the sense of LMS methods, the entire family results in energy and momentum conserving algorithms regardless of the selection of ρ_3∞ since only displacement and velocity participate in the algorithm as clearly presented in this paper. Numerious numerical illustration are presented to demonstrate the fundamental principles.