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自启动单解显式时间积分器的研究进展及性能评估

李金泽 刘耀坤 于开平 崔乃刚

李金泽, 刘耀坤, 于开平, 崔乃刚. 自启动单解显式时间积分器的研究进展及性能评估. 力学进展, 2025, 55(2): 1-52 doi: 10.6052/1000-0992-24-030
引用本文: 李金泽, 刘耀坤, 于开平, 崔乃刚. 自启动单解显式时间积分器的研究进展及性能评估. 力学进展, 2025, 55(2): 1-52 doi: 10.6052/1000-0992-24-030
Li J Z, Liu Y K, Yu K P, Cui N G. Research progress and performance evaluations for self-starting single-solve explicit time integrators. Advances in Mechanics, 2025, 55(2): 1-52 doi: 10.6052/1000-0992-24-030
Citation: Li J Z, Liu Y K, Yu K P, Cui N G. Research progress and performance evaluations for self-starting single-solve explicit time integrators. Advances in Mechanics, 2025, 55(2): 1-52 doi: 10.6052/1000-0992-24-030

自启动单解显式时间积分器的研究进展及性能评估

doi: 10.6052/1000-0992-24-030 cstr: 32046.14.1000-0992-24-030
基金项目: 国家自然科学基金项目(12272105), 国家资助博士后研究人员计划(GZC20233464), 中国博士后科学基金资助项目(2024M764165), 黑龙江省博士后资助项目(LBH-Z23153). 作者非常感谢两位审稿人细致认真地评阅稿件并给出了非常宝贵的建议
详细信息
    作者简介:

    于开平, 博士生导师, 哈尔滨工业大学航天学院力学学科长聘教授、结构动力学团队负责人. 主要从事航天器结构动力学设计与反问题、振动控制、结构动强度、人工智能在振动工程中的应用等研究, 涵盖热/力/振/噪多场耦合、非线性气动弹性、中高频振动与噪声、动力学先进数值算法、载荷与参数辨识等方向. 入选教育部“新世纪优秀人才”计划、黑龙省教学名师, 国家重大科技工程专项论证专家组成员. 主持国家部委基础研究重点、国家重大科技工程、国家自然科学基金等国家级课题二十余项, 发表学术论文300余篇. 目前任中国振动工程学会常务理事、模态分析与试验专委会主任、教育工作委员会副主任, 中国力学学会动力学与控制专委会委员、教育工作委员会委员

    通讯作者:

    yukp@hit.edu.cn

  • 中图分类号: O302

Research progress and performance evaluations for self-starting single-solve explicit time integrators

More Information
  • 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.

     

  • 图  1  EWBZ-$\alpha $法的谱半径变化曲线. (a)在${\rho _{\rm{b}}} = 0.4$$\xi = 0$时谱半径随${\rho _{\rm{s}}}$变化; (b)在$\xi = 0$${\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  3  振幅与相位误差分析流程

    图  4  最优EWBZ-$\alpha $法在$\xi = 0$时的相对周期误差(PE)

    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$${\rho _{\rm{s}}} = 0$; (b)${\rho _{\rm{b}}} = 0$${\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  14  GSSE法(${\rho _{\rm{b}}} = 0.9$)在取$\Delta t = 0.02$ s求解无阻尼自由振动时预测的数值位移

    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$A = 1\;{{\text{m}}^2}$, $E = 3 \times {10^7}\;{\text{Pa}}$, $L = 20\;{\text{m}}$, $\rho = 7.4 \times {10^{ - 4}}\;{\text{kg}}/{{\text{m}}^3}$

    图  18  使用对角质量矩阵$(r = 0)$得到的数值解

    图  19  使用高阶质量矩阵$(r = 1/2)$得到的数值解

    图  20  中心受力的方形薄膜

    图  21  不同算法计算得到的二维波传导问题 (a) $\theta = 0$; (b) $\theta = \pi /4$

    图  22  不同算法计算得到的位移等值线图

    图  23  不同算法计算得到的速度等值线图

    图  24  不同算法计算得到的加速度等值线图

    图  25  平面弹簧摆

    图  26  不同算法计算弹簧摆得到的水平位移曲线

    图  27  间隙非线性的折叠舵面模型. (a)示意图, (b)有限元模型

    图  28  各显式算法求解间隙非线性折叠舵面的位移时域曲线

    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$\varsigma = 2{\rho _{\rm{b}}}/({\rho _{\rm{b}}} + 1)$.
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    下载: 导出CSV

    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 ${\rho _{\rm{b}}} = 1$
    下载: 导出CSV

    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
    下载: 导出CSV

    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
    下载: 导出CSV
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  • 收稿日期:  2024-09-20
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