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.

中文翻译：

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关于计算动力学的新型零阶过冲 LMS 算法


在本文中，在结构动力学问题的双场形式中开发了一种新的时间加权残差方法，以通过设计实现广义类最优零阶过冲线性多步 （LMS） 算法。我们首次在二阶瞬态系统的双场形式中开发了一种新的时间加权残差方法，从而产生了新提出的 ZOO4 方案（具有 4 个根的零阶过冲）以实现：位移、速度和加速度的二阶时间精度、无条件稳定性、零阶过冲、可控的数值耗散/色散和最小的计算复杂性。特别是，它解决了现有单步方法中的问题，这些问题在位移和/或速度上表现出一阶过冲。此外，对新提出的 ZOO4 方案与现有方法之间的关系进行了对比和分析，从时间加权残差的角度为最近的文献进展提供了新的和深入的理解。通过各种数值示例进行了严格的数值分析、验证和确认，以证实所提出的方法在准确性和稳定性分析中的重要性，特别是展示了在线性/非线性结构动力学问题的数值耗散方案中实现零阶过冲的进步。