瞬态动力学直接自启动子步隐式积分器的阶数提升
Order elevation of directly self-starting sub-step implicit integrators for transient dynamics
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中文总结 AI 辅助
本研究提出广义s子步隐式框架，通过释放子步位置约束实现阶数提升，在不增加子步数下达到七阶精度，并验证了可控耗散与超收敛性。
中文摘要 AI 辅助
直接自启动隐式方法因避免了辅助启动过程，同时保留原始一阶或二阶控制方程，在瞬态分析中具有吸引力。然而，大多数现有公式通常将最后一个子步固定在每个时间区间的末端，这限制了可达到的阶数。本研究通过释放这一约束并将所有子步位置视为设计变量，发展了一个广义的$s$-子步隐式框架。所得方法对一阶和二阶瞬态系统均允许统一的Runge--Kutta表示，并在所有子步上保持相同的有效矩阵。通过同时匹配数值放大因子和载荷算子来推导精度条件，从而同时考虑齐次响应和强迫响应。对于$s=1,\cdots,6$，获得了两个互补的族：具有用户可控高频数值耗散和可调子步位置的$s$阶成员，以及通过选择子步位置获得的$(s+1)$阶成员，其耗散固定。后者在不增加子步数量的情况下达到高达七阶精度，尽管一些高阶成员是$A(\alpha)$-稳定的，其稳定角极其接近$90^\circ$。解析幅值和相位误差进一步揭示了无阻尼系统中的奇偶依赖超收敛性，适当的参数选择可以大幅提高相位或幅值精度，超越形式阶数。数值基准验证了预测的收敛阶数以及对虚假高频响应的可控抑制。
英文摘要
Directly self-starting implicit methods are attractive for transient analysis because they avoid auxiliary starting procedures while retaining the original first- or second-order governing equations. However, most existing formulations usually fix the last sub-step at the end of each time interval, which restricts the attainable order. This study develops a generalized $s$-sub-step implicit framework by releasing this constraint and treating all sub-step locations as design variables. The resulting methods admit a unified Runge--Kutta representation for both first- and second-order transient systems and preserve identical effective matrices over all sub-steps. Accuracy conditions are derived by simultaneously matching the numerical amplification factor and load operator, thereby accounting for both homogeneous and forced responses. For $s=1,~\cdots,~6$, two complementary families are obtained: $s$th-order members with user-controllable high-frequency numerical dissipation and adjustable sub-step locations, and $(s+1)$th-order members obtained by selecting the sub-step locations, with fixed dissipation. The latter reach up to seventh-order accuracy without increasing the number of sub-steps, although some high-order members are $A(α)$-stable with stability angles extremely close to $90^\circ$. Analytical amplitude and phase errors further reveal parity-dependent superconvergence in undamped systems, and appropriate parameter selections can substantially increase either phase or amplitude accuracy beyond the formal order. Numerical benchmarks confirm the predicted convergence orders and the controllable suppression of spurious high-frequency responses.
发表机构
- Harbin Institute of Technology（哈尔滨工业大学）
- Zhongke Ruilong Intelligent Manufacturing (Jiangsu) Co., Ltd.（中科瑞龙智能制造有限公司）
- Nanyang Technological University（南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。
