On the novel zero-order overshooting LMS algorithms by design for computational dynamics

In this paper, a novel time-weighted residual methodology is developed in the two-field form of structural dynamics problems to enable generalized class of optimal zero-order overshooting Linear Multi-Step (LMS) algorithms by design. For the first time, we develop a novel time-weighted residual methodology in the two-field form of the second-order time-dependent systems, leading to the newly proposed ZOO4 schemes (zero-order overshooting with 4 roots) to achieve: second-order time accuracy in displacement, velocity, and acceleration, unconditional stability, zero-order overshooting, controllable numerical dissipation/dispersion, and minimal computational complexity. Particularly, it resolves the issues in existing single-step methods, which exhibit first-order overshooting in displacement and/or velocity. Additionally, the relationship between the newly proposed ZOO4 schemes and existing methods is contrasted and analyzed, providing a new and in-depth understanding to the recent advances in literature from the time-weighted residual viewpoint. Rigorous numerical analysis, verification, and validation via various numerical examples are presented to substantiate the significance of the proposed methodology in accuracy and stability analysis, particularly demonstrating the advancements towards achieving zero-order overshooting in numerically dissipative schemes for linear/nonlinear structural dynamics problems. © 2024 The Authors