Research progress and performance evaluations for self-starting single-solve explicit time integrators
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摘要: 直接时间积分法在大型非线性动力系统的数值计算中起着关键作用, 尤其是在工程仿真与设计领域. 自启动单解显式时间积分法因在处理复杂非线性系统时的高效性和可靠性, 成为该领域的重要工具. 然而, 随着此类算法的逐步发展和多样化, 性能表现各异, 因此对其进行系统回顾和深入分析具有急迫性和重要性. 本文首先介绍了评估时间积分法性能的主要指标, 包括精度、稳定性、振幅误差与相位误差等, 为读者提供了理论基础; 接着, 详细回顾了自启动单解显式时间积分法的发展历程, 系统梳理了各种算法的演变过程; 比较了多种自启动单解显式时间积分法在谱特性、精度、稳定性以及误差方面的性能表现并使用典型算例与工程结构进行了数值验证. 重点指出当前性能表现优异的两种显式积分法: 完全显式的GSSE法和速度隐式处理的GSSI法. 这两种方法都具备自启动、单解、显式求解、最大化条件稳定域、可控数值耗散(全历程变化)和一致二阶精度的特点, 而二者之间的区别在于求解阻尼问题的计算量和有阻尼时的条件稳定域大小. 本文还展望了显式时间积分法未来的研究方向, 强调了进一步优化与发展的潜力.Abstract: Direct time integration methods play a critical role in the numerical computation of large-scale nonlinear dynamic systems, particularly in the field of engineering simulation and design. The self-starting single-solve explicit time integration methods have become essential tools in this domain due to their efficiency and reliability in handling complex nonlinear systems. However, as these algorithms continue to evolve and diversify, their performance varies significantly, underscoring the urgent need for a systematic review and in-depth analysis of their capabilities. This paper first introduces the key performance metrics for evaluating time integration methods, including accuracy, stability, amplitude and phase error, providing a theoretical foundation for readers. It then offers a detailed review of the development of self-starting single-solve explicit time integration methods, systematically tracing the evolution of various algorithms. Finally, the performance of several self-starting single-solve explicit methods is compared in terms of spectral properties, accuracy, stability, and error characteristics, with numerical verification performed using typical examples and engineering structures. The paper highlights two explicit methods that currently exhibit superior performance: The fully explicit GSSE method and the velocity-implicit GSSI method. Both methods are characterized by their self-starting capability, single-solution, explicitness, maximized conditional stability, controllable numerical dissipation (over the full range), and identical second-order accuracy. The primary distinction between the two lies in the computational effort required for damping problems and the size of the conditional stability domain in the presence of damping. The paper also explores future research directions for explicit time integration methods, emphasizing the potential for further optimization and development.
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图 1 EWBZ-
$\alpha $ 法的谱半径变化曲线. (a)在${\rho _{\rm{b}}} = 0.4$ 且$\xi = 0$ 时谱半径随${\rho _{\rm{s}}}$ 变化; (b)在$\xi = 0$ 且${\rho _{\rm{b}}} = {\rho _{\rm{s}}}$ 时谱半径随${\rho _{\rm{b}}}$ 变化
图 2 EWBZ-
$\alpha $ 法的谱半径与条件稳定域. (a)在$\xi = 0.1$ 时谱半径的变化情况; (b)最优EWBZ-$\alpha $ 法的稳定域${\varOmega _{\rm{s}}}$ 的变化情况
图 5 取步长
$\Delta t = 0.02$ s, 最优EWBZ-$\alpha $ 法在求解无阻尼自由振动的单自由度系统$ (\ddot {\boldsymbol{u}}(t) + 4{\boldsymbol{u}}(t) = 0, $ $ {\text{ }}{\boldsymbol{u}}(t) = 1,\;\dot {\boldsymbol{u}}(t) $ $ = 0) $ 时预测的数值解与精确解的比较
图 6 自启动单解显式算法(
${\rho _{\rm{b}}} = {\rho _{\rm{s}}} = 1$ )在求解无阻尼自由振动时的收敛率. (a)全局位移误差; (b)全局速度误差; (c)全局加速度误差
图 7 自启动单解显式算法(
${\rho _{\rm{b}}} = {\rho _{\rm{s}}} = 0.6$ )在求解无阻尼自由振动时的收敛率. (a)全局位移误差; (b)全局速度误差; (c)全局加速度误差
图 8 自启动单解显式算法(
${\rho _{\rm{b}}} = {\rho _{\rm{s}}} = 1$ )在求解无阻尼受迫振动时的收敛率. (a)全局位移误差; (b)全局速度误差; (c)全局加速度误差
图 9 自启动单解显式算法(
${\rho _{\rm{b}}} = {\rho _{\rm{s}}} = 0.6$ )在求解无阻尼受迫振动时的收敛率. (a)全局位移误差; (b)全局速度误差; (c)全局加速度误差
图 10 自启动单解显式算法(
${\rho _{\rm{b}}} = 0.6$ )在求解有阻尼自由振动时的收敛率. (a)全局位移误差; (b)全局速度误差; (c)全局加速度误差
图 11 自启动单解显式算法(
${\rho _{\rm{b}}} = 0.6$ )在求解有阻尼受迫振动时的收敛率. (a) 全局位移误差; (b) 全局速度误差; (c) 全局加速度误差
图 12 GSSE和GSSI法的谱半径. (a)
${\rho _{\rm{b}}} = 1$ 和${\rho _{\rm{s}}} = 0$ ; (b)${\rho _{\rm{b}}} = 0$ 和${\rho _{\rm{s}}} = 0$ 
图 13 GSSE和GSSI法的稳定域曲面. (a) GSSE法(
${\rho _{\rm{s}}} = {\rho _{\rm{b}}}$ ); (b) GSSE法(${\rho _{\rm{s}}} = 0$ ); (c) GSSI法(${\rho _{\rm{s}}} = {\rho _{\rm{b}}}$ ); (d) GSSI法(${\rho _{\rm{s}}} = 0$ )
图 15 GSSE和GSSI法振幅和相位误差的比较. (a) GSSE法(
${\rho _{\rm{s}}} = {\rho _{\rm{b}}}$ )的振幅误差主项; (b) GSSE法(${\rho _{\rm{s}}} = {\rho _{\rm{b}}}$ )的相位误差主项; (c) GSSI法(${\rho _{\rm{s}}} = {\rho _{\rm{b}}}$ )的振幅误差主项; (d) GSSI法(${\rho _{\rm{s}}} = {\rho _{\rm{b}}}$ )的相位误差主项
图 16 GSSE和GSSI法(
${\rho _{\rm{b}}} = {\rho _{\rm{s}}}$ )在取$\Delta t = 0.1$ s解阻尼自由振动问题时预测的数值位移. (a) GSSE法; (b) GSSI法
图 17 受冲击的直杆, 截面积
$A = 1\;{{\text{m}}^2}$ ,$E = 3 \times {10^7}\;{\text{Pa}}$ ,$L = 20\;{\text{m}}$ ,$\rho = 7.4 \times {10^{ - 4}}\;{\text{kg}}/{{\text{m}}^3}$ 表 1 自启动单解显式时间积分法的算法结构特点比较
算法 平衡方程满足的时刻 启动条件 耗散控制 速度处理方式 EN-$\beta $(Hughes & Liu 1978) ${t_{n + 1}}$ ${{\boldsymbol{\ddot u}}_0}$ $0 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 TW (Mahéo et al. 2009) ${t_{n + 1}}$ ${{\boldsymbol{\ddot u}}_0}$ $0 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 EDV1 (Zhang & Xing 2019) ${t_{n + \zeta }}$ — $0 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 TSSE (Li & Yu 2021) ${t_{n + \zeta }}$ — $0 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 EHHT-$\alpha $(Miranda et al. 1989) ${t_{n + \varsigma }}$ ${{\boldsymbol{\ddot u}}_0}$ $1/2 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 CL (Chung & Lee 1994) ${t_n}$ — $1/2 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 NT (Namburu & Tamma 1992) $ {t_{n + 1/2}} $ — $\rho_{\mathrm{b}}=1 $ 隐式 EWBZ-$\alpha $(Hulbert & Chung 1996) ${t_n}$ — $0 \leqslant \left| {{\rho _{\rm{s}}}} \right| \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 YZ (于开平 等 2004) ${t_n}$ — $0 \leqslant \left| {{\rho _{\rm{s}}}} \right| \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 EG-$\alpha $(Li et al. 2021a) ${t_{n + 2/3 - {\alpha _m}}}$ ${{\boldsymbol{\ddot u}}_0}$ $0 \leqslant \left| {{\rho _{\rm{s}}}} \right| \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 E-GSSSS (Shimada 2013) ${t_{n + {W_1}{\Lambda _1}}}$ ${{\boldsymbol{\ddot u}}_0}$ $0 \leqslant \left| {{\rho _{3{\rm{b}}}}} \right| \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 ICL (Kim 2019) ${t_{n + 1}}$ ${{\boldsymbol{\ddot u}}_0}$ $1/2 \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 NE (Newmark 1959) ${t_{n + 1}}$ ${{\boldsymbol{\ddot u}}_0}$ $\rho_{\mathrm{b}}=1 $ 隐式 GSSE (Li et al. 2021a) ${t_{n + p}}$ ${{\boldsymbol{\ddot u}}_0}$ $0 \leqslant \left| {{\rho _{\rm{s}}}} \right| \leqslant {\rho _{\rm{b}}} \leqslant 1$ 显式 GSSI (Zhao et al. 2023) ${t_{n + p}}$ ${{\boldsymbol{\ddot u}}_0}$ $0 \leqslant \left| {{\rho _{\rm{s}}}} \right| \leqslant {\rho _{\rm{b}}} \leqslant 1$ 隐式 注: $\zeta = (3 - {\rho _{\rm{b}}})/(2{\rho _{\rm{b}}} + 2)$和$\varsigma = 2{\rho _{\rm{b}}}/({\rho _{\rm{b}}} + 1)$. 
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表 2 自启动单解显式积分法在求解无阻尼问题
$(\xi = 0)$ 时的精度阶数算法 位移 速度 加速度 $f(t) = 0$ $f(t)\ne 0$ $f(t) = 0$ $f(t)\ne 0$ $f(t) = 0$ $f(t)\ne 0$ ${\rho _{\rm{b}}} = 1$ ${\rho _{\rm{b}}} \ne 1$ ${\rho _{\rm{b}}} = 1$ ${\rho _{\rm{b}}} \ne 1$ ${\rho _{\rm{b}}} = 1$ ${\rho _{\rm{b}}} \ne 1$ ${\rho _{\rm{b}}} = 1$ ${\rho _{\rm{b}}} \ne 1$ ${\rho _{\rm{b}}} = 1$ ${\rho _{\rm{b}}} \ne 1$ ${\rho _{\rm{b}}} = 1$ ${\rho _{\rm{b}}} \ne 1$ TW 1 1 1 1 2 1 1 1 1 1 1 1 EN-$\beta $ 2 1 2 1 2 1 2 1 2 1 2 1 EDV1 2 1 2 1 2 1 2 1 — — — — TSSE 2 1 2 1 2 1 2 1 1 1 1 1 EHHT-$\alpha $ 2 2 2 2 2 1 2 1 2 1 2 1 CL 2 2 2 2 2 2 2 2 1 1 1 1 NT 2 — 2 — 2 — 2 — 1 — 1 — EWBZ-$\alpha $ 2 2 2 2 2 2 2 2 1 1 1 1 YZ 2 2 2 2 2 2 2 2 1 1 1 1 EG-$\alpha $ 2 2 2 2 2 2 2 2 1 1 1 1 ICL 2 2 2 2 2 2 2 2 2 2 2 2 NE 2 — 2 — 2 — 2 — 2 — 2 — GSSE 2 2 2 2 2 2 2 2 2 2 2 2 GSSI 2 2 2 2 2 2 2 2 2 2 2 2
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表 3 自启动单解显式算法在求解有阻尼问题
$(\xi \ne 0)$ 时的精度阶数算法 位移 速度 加速度 $f(t) = 0$ $f(t)\ne 0$ $f(t) = 0$ $f(t)\ne 0$ $f(t) = 0$ $f(t)\ne 0$ EN-$\beta $ 1 1 1 1 1 1 TW 1 1 1 1 1 1 EDV1 1 1 1 1 — — TSSE 1 1 1 1 1 1 EHHT-$\alpha $ 1 1 1 1 1 1 CL 2 2 2 2 1 1 NT 2 2 2 2 1 1 EWBZ-$\alpha $ 2 2 2 2 1 1 YZ 2 2 2 2 1 1 EG-$\alpha $ 2 2 2 2 1 1 ICL 2 2 2 2 2 2 NE 2 2 2 2 2 2 GSSE 2 2 2 2 2 2 GSSI 2 2 2 2 2 2
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表 4 自启动单解显式积分法的条件稳定域
算法 稳定域($\xi \ne 0$) 稳定域($\xi = 0$) 最大值 当$\xi $增大时 最大值 参数取值 EN-$\beta $、TW、EDV1、TSSE <2 减小 2 ${\rho _{\rm{b}}} = 1$ EHHT-$\alpha $ <2 减小 2 ${\rho _{\rm{b}}} = 1$ CL、ICL <2 减小 2 ${\rho _{\rm{b}}} = 1$ NT、NE 2 不变 2 — EWBZ-$\alpha $、YZ、EG-$\alpha $ <2 减小 2 ${\rho _{\rm{b}}} = 1$ GSSE <2 减小 2 ${\rho _{\rm{b}}} = 1$ GSSI <3.8271 增大 2 ${\rho _{\rm{b}}} = 1$
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表 5 自启动单解显式积分算法的振幅与相位误差阶数
算法 有阻尼($\xi \ne 0$) 无阻尼($\xi = 0$) 振幅误差 相位误差 振幅误差 相位误差 EN-$\beta $、TW、EDV1、TSSE 1 1 1 2 EHHT-$\alpha $ 1 1 3 2 CL、ICL 2 2 3 2 NT、NE 2 2 — 2 EWBZ-$\alpha $、EG-$\alpha $、GSSE、YZ 2 2 3 2 GSSI 2 2 3 2
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表 6 不同算法计算间隙非线性舵面的性能比较
算法 参数取值 全局误差($ \times {10^{ - 4}}$) 运行时间(s) EN-$\beta $(Hughes & Liu 1978) ${\rho _{\rm{b}}} = 1,\beta = 0$ 2 094.65400691130 16.7371675 TW (Mahéo et al. 2009) ${\rho _{\rm{b}}} = 1$ 2 094.65400539673 16.5674646 TSSE (Li & Yu 2021) ${\rho _{\rm{b}}} = 1$ 2 095.11359385971 16.6394112 EHHT-$\alpha $(Miranda et al. 1989) ${\rho _{\rm{b}}} = 1$ 2 094.64833439895 16.7600374 CL (Chung & Lee 1994) ${\rho _{\rm{b}}} = 1$ 122.05490051270 16.5825193 NT (Namburu & Tamma 1992) — 128.74544603336 16.6789087 EWBZ-$\alpha $(Hulbert & Chung 1996) ${\rho _{\rm{b}}} = 1,{\rho _{\rm{s}}} = 0$ 122.05490051270 16.7971924 ${\rho _{\rm{b}}} = 0,{\rho _{\rm{s}}} = 0$ 574.57122300857 16.9433569 ${\rho _{\rm{b}}} = 0.36653,{\rho _{\rm{s}}} = 0.36653$ 8.09663420031 17.0288960 EG-$\alpha $(Li et al. 2021a) ${\rho _{\rm{b}}} = 1,{\rho _{\rm{s}}} = 0$ 120.23431346855 17.0691251 ${\rho _{\rm{b}}} = 0,{\rho _{\rm{s}}} = 0$ 579.63193462968 17.1157046 ${\rho _{\rm{b}}} = 0.36653,{\rho _{\rm{s}}} = 0.36653$ 9.67169868571 16.7288852 ICL (Kim 2019) ${\rho _{\rm{b}}} = 1$ 6.72052405612 16.6451444 NE (Newmark 1959) — 128.86982904751 16.5817191 CD (Fox & Goodwin 1949) — 128.89359314558 16.8137596 GSSE (Li et al. 2021a) ${\rho _{\rm{b}}} = 1,{\rho _{\rm{s}}} = 0$ 122.05489777818 17.0365016 ${\rho _{\rm{b}}} = 0,{\rho _{\rm{s}}} = 0$ 577.66327372322 16.5218805 ${\rho _{\rm{b}}} = 0.36653,{\rho _{\rm{s}}} = 0.36653$ 7.02558590034 16.7662669 GSSI (Zhao et al. 2023) ${\rho _{\rm{b}}} = 1,{\rho _{\rm{s}}} = 0$ 128.86982904751 16.7886440 ${\rho _{\rm{b}}} = 0,{\rho _{\rm{s}}} = 0$ 555.75987795750 16.6073923 ${\rho _{\rm{b}}} = 0.36653,{\rho _{\rm{s}}} = 0.36653$ 2.79079214905 16.6471036 TPO/G-$\alpha $
(Chung & Hulbert 1993, 邵慧萍和蔡承文 1988)${\rho _\infty } = 1$ 233.37572689655 32.7353195 ${\rho _\infty } = 0$ 1091.11262975068 32.8101065 Newmark-$\beta $(Newmark 1959) $\gamma = 0.5,\beta = 0.25$ 233.56743425553 33.1281854 SUCI2 (Li & Yu 2020) ${\rho _\infty } = 1$ 66.39354082933 122.886593 S2E1 (Li & Yu 2021) ${\rho _{\rm{b}}} = 0.99$ 30.98184714029 33.2407046
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