结 构 动 力 学 自 启 动 单 解 隐 式 积 分 器 的 分 析 与 优 化

Time integration algorithms are widely used and efficient numerical techniques for analyzing the dynamic response of large aerospace structures. Among the various algorithms, competitive implicit time integration methods generally need to meet a series of stringent requirements, including self-starting, single-solve, unconditional stability, second-order accuracy, controllable numerical high-frequency dissipation, and zero-order overshoot. To further investigate the limitations of self-starting and single-solve implicit integration methods, this paper constructs a universally applicable self-starting and single-solve time integration algorithm framework and provides a detailed analysis of the algebraic conditions required to achieve identical accuracy orders, overshoot orders, and controllable numerical high-frequency dissipation. Based on this framework, it is proven that, without additional techniques, self-starting and single-solve implicit integration methods cannot simultaneously achieve second-order accuracy for both displacement and velocity, zero-order overshoot in displacement, at most first-order overshoot in velocity, and controllable numerical high-frequency dissipation. Given this limitation, the paper further designs, analyzes, and optimizes three classes of implicit integration algorithms with different overshoot orders. The new algorithms all possess self-starting, single-solve, unconditional stability, second-order accuracy, and controllable numerical high-frequency dissipation (a full range from zero to one), with the differences lying in the accuracy order of acceleration and the order of overshoot. Through optimization of amplitude and phase errors, the three new algorithms outperform the previously published self-starting and single-solve implicit integration methods. Performance comparisons and numerical examples validate the theoretical analysis and clearly demonstrate the significant advantages of the three new algorithms in terms of accuracy.