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Article(id=1241794071639953456, tenantId=1146029695717560320, journalId=1241701559352995854, issueId=1241794070289387562, articleNumber=null, orderNo=null, doi=10.13197/j.eeed.2025.0515, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1711123200000, receivedDateStr=2024-03-23, revisedDate=1715184000000, revisedDateStr=2024-05-09, acceptedDate=null, acceptedDateStr=null, onlineDate=1773996378097, onlineDateStr=2026-03-20, pubDate=1761062400000, pubDateStr=2025-10-22, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773996378097, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773996378097, creator=13701087609, updateTime=1773996378097, updator=13701087609, issue=Issue{id=1241794070289387562, tenantId=1146029695717560320, journalId=1241701559352995854, year='2025', volume='45', issue='5', pageStart='1', pageEnd='227', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=1, specialIssue=null, createTime=1773996377775, creator=13701087609, updateTime=1773996935444, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241796409465307627, tenantId=1146029695717560320, journalId=1241701559352995854, issueId=1241794070289387562, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241796409465307628, tenantId=1146029695717560320, journalId=1241701559352995854, issueId=1241794070289387562, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=154, endPage=163, ext={EN=ArticleExt(id=1241794071900000310, articleId=1241794071639953456, tenantId=1146029695717560320, journalId=1241701559352995854, language=EN, title=GPU accelerated explicit non-linear mode superposition method for real-time analysis of large-scale structures, columnId=1241794071602200899, journalTitle=Earthquake Engineering and Engineering Dynamics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

In this paper, a GPU-accelerated explicit nonlinear mode superposition method (ENMS) is proposed for real-time computation of large-scale bridge structures with local nonlinearities, especially non-linear dampers. This method treats the non-linear damping force as an external load, decouples linear equations of motion at each discretized time step by using the mode superposition method, and solves them by using an explicit step-by-step integration method. In this way, this method avoids the iterative solution to the equations of motion, and significantly improves computational efficiency with the fast mode superposition method. In view of the decoupled equations, a GPU is utilized to accelerate the computation, thereby further improving the computational efficiency. Numerical simulation studies on a large-span cable-stayed bridge show the followings: For a large multi-degree-of-freedom structure with local nonlinearities, this method can be used to conveniently and accurately solve dynamic responses with the parameters exported from Midas Civil. For an exponential Maxwell model of viscous dampers, the dichotomous method is able to accurately solve the damping force, providing a better solution to the problem of modeling non-linear dampers. The GPU acceleration can significantly improve the computational efficiency of the explicit non-linear mode superposition method.

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该文针对具有局部耗能非线性的大型桥梁结构的实时计算问题,提出了一种图形处理器（graphics processing unit, GPU）加速的显式非线性振型叠加法（explicit nonlinear mode superposition method, ENMS）。该方法视非线性反力为外荷载,采用振型叠加法对单积分步线性运动方程进行解耦,并采用显式逐步积分法求解,避免了运动方程迭代求解,利用了振型叠加法的快速计算优势,大幅提升计算效率。针对方程已经解耦的特征,利用GPU加速计算,进一步提升计算效率。某大跨度斜拉桥数值仿真研究表明：对于存在局部非线性的大型多自由度结构,该方法可利用Midas Civil导出的参数方便地进行动力响应求解,结果精度高；对于黏滞阻尼器的指数阻尼Maxwell模型,二分法能够准确求解阻尼力,较好地解决非线性阻尼器建模问题；GPU加速可显著提升显式非线性振型叠加法计算效率。

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吴斌（1970—）,男,教授,博士,主要从事结构试验方法、振动控制研究。E-mail：
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王贞（1983—）,男,研究员,博士,主要从事结构混合试验方面研究。E-mail：

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王贞（1983—）,男,研究员,博士,主要从事结构混合试验方面研究。E-mail：

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Problems and solutions of GPU parallel computing in Matlab[J]. Computer & Network, 2013, 39(11): 51-54. 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Application of GPU parallel computing method based on MATLAB[J]. Automation & Instrumentation, 2017(6): 172-174. 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Configuration parameters

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硬件型号
CPU处理器Intel（R）Core（TM）i7-10700F CPU@2.90GHz
GPU处理器NVIDIA GeForce GTX 1660 SUPER
内存16.0 GB
操作系统Windows 11专业版64位
CUDA版本CUDA Toolkit 11.8
MATLAB版本R2019a
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配置参数表

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硬件型号
CPU处理器Intel（R）Core（TM）i7-10700F CPU@2.90GHz
GPU处理器NVIDIA GeForce GTX 1660 SUPER
内存16.0 GB
操作系统Windows 11专业版64位
CUDA版本CUDA Toolkit 11.8
MATLAB版本R2019a
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Computational efficiency of bridge models under different methods

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方法平均耗时
CPU计算GPU加速计算加速比RG
直接求解124.50 ms65.10 ms1.91
ENMS法44.30 ms2.00 ms22.15
加速比RM2.8132.5562.25
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不同方法下桥梁模型的计算效率

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方法平均耗时
CPU计算GPU加速计算加速比RG
直接求解124.50 ms65.10 ms1.91
ENMS法44.30 ms2.00 ms22.15
加速比RM2.8132.5562.25
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Efficiency of GPU acceleration under different computing scales

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自由度数直接求解ENMS法
TCPU/ms TGPU/ms加速比RG TCPU/ms TGPU/ms加速比RG
5003.34.50.71.80.82.3
100014.59.71.58.90.99.9
200093.043.52.135.72.017.9
3000211.1126.91.779.94.119.5
5000693.5552.51.3211.410.120.9
70001951.61444.91.4406.019.221.1
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不同计算规模下GPU加速效率

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自由度数直接求解ENMS法
TCPU/ms TGPU/ms加速比RG TCPU/ms TGPU/ms加速比RG
5003.34.50.71.80.82.3
100014.59.71.58.90.99.9
200093.043.52.135.72.017.9
3000211.1126.91.779.94.119.5
5000693.5552.51.3211.410.120.9
70001951.61444.91.4406.019.221.1
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Efficiency of ENMS method under different computing scales

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自由度数CPU计算GPU加速计算
T/ms TM/ms加速比RM T/ms TM/ms加速比RM
5003.31.81.84.50.85.6
100014.58.91.69.70.910.8
200093.035.72.643.52.021.8
3000211.179.92.6126.94.131.0
5000693.5211.43.3552.510.154.7
70001951.6406.04.81444.919.275.3
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不同计算规模下ENMS法效率

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自由度数CPU计算GPU加速计算
T/ms TM/ms加速比RM T/ms TM/ms加速比RM
5003.31.81.84.50.85.6
100014.58.91.69.70.910.8
200093.035.72.643.52.021.8
3000211.179.92.6126.94.131.0
5000693.5211.43.3552.510.154.7
70001951.6406.04.81444.919.275.3
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面向大型结构实时计算的显卡加速显式非线性振型叠加法
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王贞 1, 2 , 黄健君 1 , 武文斌 1 , 吴斌 1, 2 , 徐小洋 3
地震工程与工程振动 | 研究论文 2025,45(5): 154-163
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地震工程与工程振动 | 研究论文 2025, 45(5): 154-163
面向大型结构实时计算的显卡加速显式非线性振型叠加法
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王贞1, 2 , 黄健君1, 武文斌1, 吴斌1, 2 , 徐小洋3
作者信息
  • 1.武汉理工大学土木工程与建筑学院,湖北武汉430070
  • 2.武汉理工大学三亚科教创新园,海南三亚572000
  • 3.中国建筑第二工程局有限公司华中公司,湖北武汉430061

    • 王贞（1983—）,男,研究员,博士,主要从事结构混合试验方面研究。E-mail：

通讯作者:

吴斌（1970—）,男,教授,博士,主要从事结构试验方法、振动控制研究。E-mail：
GPU accelerated explicit non-linear mode superposition method for real-time analysis of large-scale structures
Zhen WANG1, 2 , Jianjun HUANG1, Wenbin WU1, Bin WU1, 2 , Xiaoyang XU3
Affiliations
  • 1.School of Civil Engineering & Architecture, Wuhan University of Technology, Wuhan 430070, China
  • 2.Sanya Science and Education Innovation Park, Wuhan University of Technology, Sanya 572000, China
  • 3.China Construction Second Engineering Bureau Co., Ltd., Central China Branch, Wuhan 430061, China
出版时间: 2025-10-22 doi: 10.13197/j.eeed.2025.0515
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摘要
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该文针对具有局部耗能非线性的大型桥梁结构的实时计算问题,提出了一种图形处理器（graphics processing unit, GPU）加速的显式非线性振型叠加法（explicit nonlinear mode superposition method, ENMS）。该方法视非线性反力为外荷载,采用振型叠加法对单积分步线性运动方程进行解耦,并采用显式逐步积分法求解,避免了运动方程迭代求解,利用了振型叠加法的快速计算优势,大幅提升计算效率。针对方程已经解耦的特征,利用GPU加速计算,进一步提升计算效率。某大跨度斜拉桥数值仿真研究表明：对于存在局部非线性的大型多自由度结构,该方法可利用Midas Civil导出的参数方便地进行动力响应求解,结果精度高；对于黏滞阻尼器的指数阻尼Maxwell模型,二分法能够准确求解阻尼力,较好地解决非线性阻尼器建模问题；GPU加速可显著提升显式非线性振型叠加法计算效率。

大型桥梁结构  /  局部非线性  /  非线性振型叠加法  /  显式积分算法  /  GPU加速计算  /  实时计算分析

In this paper, a GPU-accelerated explicit nonlinear mode superposition method (ENMS) is proposed for real-time computation of large-scale bridge structures with local nonlinearities, especially non-linear dampers. This method treats the non-linear damping force as an external load, decouples linear equations of motion at each discretized time step by using the mode superposition method, and solves them by using an explicit step-by-step integration method. In this way, this method avoids the iterative solution to the equations of motion, and significantly improves computational efficiency with the fast mode superposition method. In view of the decoupled equations, a GPU is utilized to accelerate the computation, thereby further improving the computational efficiency. Numerical simulation studies on a large-span cable-stayed bridge show the followings: For a large multi-degree-of-freedom structure with local nonlinearities, this method can be used to conveniently and accurately solve dynamic responses with the parameters exported from Midas Civil. For an exponential Maxwell model of viscous dampers, the dichotomous method is able to accurately solve the damping force, providing a better solution to the problem of modeling non-linear dampers. The GPU acceleration can significantly improve the computational efficiency of the explicit non-linear mode superposition method.

large-scale bridge structure  /  local nonlinearity  /  nonlinear mode superposition method  /  explicit integration algorithms  /  GPU accelerated computation  /  real-time analysis
王贞, 黄健君, 武文斌, 吴斌, 徐小洋. 面向大型结构实时计算的显卡加速显式非线性振型叠加法. 地震工程与工程振动, 2025 , 45 (5) : 154 -163 . DOI: 10.13197/j.eeed.2025.0515
Zhen WANG, Jianjun HUANG, Wenbin WU, Bin WU, Xiaoyang XU. GPU accelerated explicit non-linear mode superposition method for real-time analysis of large-scale structures[J]. Earthquake Engineering and Engineering Dynamics, 2025 , 45 (5) : 154 -163 . DOI: 10.13197/j.eeed.2025.0515
正文
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0　引言
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桥梁作为交通生命线的关键节点,其结构安全性能对抗震救灾、灾后重建工作意义非凡,得到了社会的高度重视。桥梁抗震设计已经成为桥梁建造中的重要一环[1]。目前,主要通过有限元数值分析[2]与桥梁抗震试验[3]研究桥梁结构抗震性能。有限元分析不能准确反映桥梁结构的受力过程,得到破坏形态与实际破坏形态有差别[4]。桥梁试验中若取非线性部分作为对象单独测试,会割断整体桥梁结构响应对非线性部分的影响,若采用缩尺模型对整体桥梁结构进行试验,试验精度会受到结构尺寸影响,由于小尺寸试件混凝土与钢筋的间隙相对足尺试件较大,当缩尺试件的尺寸过小时可能导致试验精度较低[5]。基于以上原因,为了有效提高试验结果准确性和节约成本,数值模拟与试验相结合的混合试验是桥梁抗震试验的合理方法。
混合试验将整体结构划分为数值子结构和试验子结构[6],数值子结构是整体结构中能准确建立动力学方程并便于计算分析的部分,通过数值积分方法求解动力响应,而试验子结构是整体结构中难以准确建模的部分,可通过试验测量来得到该部分的动力响应。在2个子结构边界上,通过加载设备保证变形协调和荷载平衡从而实现界面耦合。大型桥梁结构的耗能部分常具有很强的非线性或速度相关性,通过实时混合试验[7]研究耗能部件及结构整体性能,是桥梁抗震试验的发展趋势。实时混合试验中要求数值子结构的计算耗时小于积分步长,由于大型桥梁结构数值模型较为复杂,计算规模较大,其实时计算问题亟待研究。NAKASHIMA等[8]完成了12自由度结构的混合试验实时计算。BLAKEBOROUGH等[9]对振型叠加法加以改进,在2001年实现了50自由度模型的实时计算。王进廷等[10]使用2台计算机分别进行数值子结构有限元计算与位移信号插值,在2014年实现了超过1 200自由度的线弹性框架-基础体系实时计算。王进廷等[11]应用多核CPU并行计算,在2018年实时求解了超过40 000自由度的线弹性基础数值结构。唐贞云等[12]应用Python-GPU并行计算,在2021年实现了超过24000自由度线弹性土体数值结构的实时计算。然而,目前对于如何快速求解包含非线性部分的数值子结构研究不足,针对考虑局部非线性的大型复杂桥梁结构,其大规模结构数值建模、实时计算问题需要进一步解决。
为了开展具有局部非线性的大跨度斜拉桥的抗震实时混合试验,考虑到常常采用Midas Civil软件完成桥梁结构建模与验算,本文直接利用该软件导出模型部分信息,解决大规模桥梁结构数值建模问题。针对大规模数值模型的实时计算问题,本文在算法上构造了显式非线性振型叠加法,在硬件上采用GPU加速方法提升计算效率。研究表明,这些措施能大幅提升计算效率,为大规模结构的实时计算及实时混合试验奠定了基础。
1　减震桥梁结构的显式非线性振型叠加法
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Midas Civil软件能够导出的结构模型参数包括：节点平动质量、结构各阶频率和振型信息等。软件无法直接导出结构的刚度矩阵和阻尼矩阵,需要通过振型叠加法利用导出的结构参数计算结构响应。尽管该思路仅适用于结构力学线性问题,该方法针对黏滞阻尼耗能的大型桥梁结构的实时分析仍然具有重要意义。这是因为：①桥梁抗震规范规定在罕遇地震下重要构件仍然处于轻微损伤工作状态[13],采用线弹性主体结构分析模型是合理的；②出现非线性力学行为的部件,如黏滞阻尼器和边辅墩等,仅仅为局部构件。因此,以整体结构地震响应、黏滞阻尼器动力响应过程为研究目标的实时混合试验,采用仅包含局部非线性的结构模型,是合理的。
不过,解决具有局部非线性的大规模结构的实时求解问题,仍然是完成大型耗能桥梁结构实时混合试验的前提。对于线性结构动力学问题,振型叠加法能够实现时程级响应解耦,可以通过筛选合适的振型以降低计算规模,能够快速简化求解大规模结构动力响应。这使得振型叠加法的运用在大型结构的实时计算中成为合适的选择。WILSON等[14]以振型叠加法为基础,针对局部非线性动力学快速求解问题,提出快速非线性分析法（fast nonlinear analysis method, FNA）[15]。本文以此为基础,针对实时混合试验对计算实时性提出的更高要求,构造了基于显式逐步积分与非线性振型叠加法的快速计算方法（explicit nonlinear mode superposition method, ENMS）,能够解决局部非线性大规模结构实时混合试验的数值建模以及快速计算问题。
1.1　基本原理
与FNA方法的思路类似,本文把结构运动方程中表现出非线性行为的局部构件的反力作为外荷载来处理。这样,局部非线性动力学问题就转化为外荷载作用下的线性动力学问题,可以采用振型叠加法或者直接积分法求解。需要说明的是,由于非线性反力与结构状态有关,并不能像线性问题的振型叠加法那样分别求得各个振型响应时程再叠加,而是需要求解结构状态与求解非线性反力交替进行,在时间轴上逐步推进。FNA采用多项式拟合离散时间步内非线性反力,并套用各项反力作用下振型响应的解析表达式直接计算,从而大幅提升了计算效果。不过,有时为了获得更高的计算精度,需要在同一时间点多次迭代求解运动方程。而运动方程迭代过程为实时混合试验的计算耗时带来不确定性,一般不建议直接采用这些方法。本文结合FNA方法的基本思想和时程分析显式方法[16],构造了适用于实时计算的新方法。
在t时刻,结构计算模型的运动平衡方程可表示为
式中：M、C、K分别为结构的质量矩阵、阻尼矩阵和弹性刚度矩阵。R（t）NL、R（t）分别为非线性单元力、外力。非线性振型叠加法的基本思想是,把R（t）NL移动到方程右端,视作外荷载的一部分。忽略非线性单元的刚度,可能导致模型不稳定。为此,可在非线性单元的位置添加任意刚度的“有效弹性单元”,同时添加相应的外荷载维持力学平衡。因此,式（1）的两边加上有效力Keu（t）,平衡方程可表示为
式中Ke为有效刚度,一般可取非线性单元的初始弹性刚度,例如阻尼器模型中弹簧部分的刚度。式（2）可进一步改写为
式中弹性刚度矩阵,有效外荷载。显然,式（3）左侧表示等效线性结构的惯性力、阻尼力和弹性恢复力,右侧包括与结构运动状态相关的非线性内力。
对应式（3）,采用振型叠加法可进一步表示为
式中M∗、K∗、C∗分别为振型质量矩阵、刚度矩阵和阻尼矩阵,前两者可表示为
式中Φ为N个关于质量矩阵、刚度矩阵正交的Ritz向量。振型荷载矩阵F∗可由式（6）求出：
显然,能对式（3）左侧解耦,因此式（4）的第n行,即结构的第n阶模态方程为
式中f（t）n为模态荷载。对于非线性单元而言,f（t）n为在同一时间点上所有模态响应的函数,因此模态方程组必须同时积分,并不能实现时程级别响应解耦。
在FNA方法中,对等效振型荷载拟合多项式函数,并采用精确积分求解模态方程。为了获得较高精度,有时需要采用运动方程迭代求解。与此不同,可采用显式逐步积分方法直接离散化求解式。以Chang方法[17-19]为例,在混合试验中相同条件下该方法的稳定性优于中心差分法。已知第i积分步结构的第n阶模态位移、速度、加速度分别为yi,n,,即可求得t+Δt时刻的模态位移yi+1,n,如式（8）所示：
式中β1和β2为算法参数,即
此后,进行振型叠加,以得到结构的物理位移u（t+Δt）。若非线性单元的反力仅与其位移相关,接着可计算其相对变形ΔuNL（t+Δt）,进而根据非线性单元的材料属性得到非线性力R（t+Δt）NL。由非线性力R（t+Δt）NL、外荷载R（t+Δt）可计算出有效外荷载,然后通过式（6）得到模态荷载F∗（t+Δt）。进而可求得该时刻的模态速度和模态加速度,如式（11）、式（12）所示：
至此,完成了该步所有必需物理量的计算,可回到式（8）开始下一步计算。该方法适用于具有局部非线性的大规模数值结构实时计算,能够通过解耦以降低计算规模,减少计算量。不过,若研究对象中大多数构件都表现出非线性,则ENMS方法无法体现其优势,不适用于该情况。
1.2　非线性黏滞阻尼器模型求解
大型桥梁通常布置阻尼器以衰减地震作用下的结构响应,其中黏滞阻尼器最为常用。黏滞阻尼器的基本原理是高压流体通过节流孔时产生黏滞阻力,是一种速度相关型阻尼器。目前有多种计算模型用于描述阻尼特性[20-21],如Kelvin模型、Maxwell模型等。Maxwell模型中阻尼单元与弹簧单元串联,见图1,能够考虑阻尼器的耗能特性和支座的变形行为,应用广泛。本文采用该模型模拟结构中所有黏滞阻尼器。
阻尼单元的指数阻尼模型可表示为
式中：cd为阻尼系数,与油缸直径、活塞直径、导杆直径和流体黏度等因素有关；s为阻尼指数,与阻尼器内部构造有关,不同的产品具有不同的取值；v0、vd分别为阻尼单元参考速度、实际速度。Maxwell模型的基本公式还包括：
式中：ks为连接弹簧刚度；d为阻尼器两端整体相对位移；ds、dd分别为弹簧段、阻尼段的相对位移。ti时刻阻尼段的速度可由其位移差分表示,即
在实时混合试验中,物理阻尼器的反力能够直接测量,数值阻尼器的反力需要通过模型求解。为了求解该非线性方程组问题,把弹簧段位移作为基本未知量,根据式（15）整理方程,可得：
显然,该式是关于弹簧段位移的非线性方程,一般可采用迭代法求解。鉴于符号函数、绝对值函数求导数的困难,本文采用二分法求解该方程。限于文章篇幅,具体细节不再赘述。
2　GPU加速计算
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采用显式非线性振型叠加法求解局部非线性大规模结构动力学问题,具有良好的计算效率,这是因为能通过筛选对结构动力响应贡献大的振型降低了问题的规模、能够实现对结构各振型的积分步解耦。而后者非常适合于并行计算,尤其是采用图形处理器计算[22-24]。为了进一步提高计算效率,满足实时混合试验的实时性要求,本文采用GPU加速计算进一步缩短计算耗时。
采用MATLAB软件作为数值子结构求解的计算软件,利用其并行计算工具箱（parallel computing toolbox）的GPU计算功能实现并行计算。需要说明的是,本文中黏滞阻尼器的反力是阻尼器两端当前位移以及上一步位移的函数,需要根据各振型的求解结果叠加获取此位移,此过程更适合在CPU计算。因此,本文的方法需要CPU计算、GPU加速相互结合。采用GPU加速计算的流程见图2。其关键步骤包括[25]：①调用gpuArray函数将变量从内存转移到显存中；②执行运算,显存中的变量会自动进行GPU计算；③调用gather函数将变量从显存转移到内存中。
实际开展实时混合实验时,要利用以上所述GPU加速计算需采用Simulink平台,并使用Real-Time Synchronization模块保证计算实时性。在Simulink平台进行GPU加速计算时需要注意以下2点：①逐步积分计算使用Simulink中的MATLAB function模块,该模块因自身特性无法调用、存储或输出GPU变量,因此,在模块内部使用持久变量,将首次转换完成的GPU变量储存,以便进行后续运算,该措施能有效提高计算效率；②由于MATLAB function模块中无法直接使用gpuArray函数命令,同时不支持GPU变量的乘法运算,因此要将计算公式写入一个m文件函数,利用coder.extrinsic命令将指定的函数声明为外部函数并调用。
为了进一步提高计算效率,对程序代码需要做适当优化。使用MATLAB/Simulink编写逐步积分计算程序时,可能会在代码中造成重复计算,产生大量冗余数据导致积分步的计算耗时较大。本文主要采取了3点措施进行优化,主要是：矩阵转换为单列向量输出；不储存可以在分析之后复现的数据；在循环外计算循环内用到的常数矩阵。限于文章篇幅,具体细节,不再赘述。
3　算例分析
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3.1　斜拉桥概况
本文以某大型斜拉桥为例,开展Midas Civil模型导出、仿真分析模型检验、计算效率比较等工作,研究所提出的实时分析方法的有效性。该桥梁立面布置见图3。该桥全长为1376 m,桥面宽为21.6 m。各跨的跨度分别为82、262、688、262、82 m,中间为1162 m采用钢箱梁,两侧边跨107 m采用结合梁,梁高为4.5 m。在主梁与桥塔横梁交界处,共布置4个纵向黏滞阻尼器,阻尼系数cd为2500 kN/（m•s-1）,阻尼指数s为0.2。
3.2　MATLAB模型正确性验证
在Midas Civil中,对该大跨斜拉桥建模并做模态分析,其中主梁、主塔、基础等均采用梁单元和集中质量模型；选取前2307个振型,导出振型信息、振型频率和节点质量等。在MATLAB中,采用显式Chang逐步积分方法,考虑GPU加速计算,编写显式非线性振型叠加法时程分析程序,得到该桥梁各个节点的位移、速度和加速度时程曲线。依据桥梁安评报告,采用50 a超越概率为10%的地震作用水平下,人工合成的地震波作为纵桥向激励,积分步长取1 ms,分析时间为30 s。本文取主梁与桥塔横梁交界处节点,比较MATLAB程序与Midas Civil时程曲线的差异,见图4。阻尼器两端的相对位移和阻尼力进行时程分析比较,见图5。
由图4可知,节点位移、速度时程结果吻合良好,加速度结果存在微小偏差。该结果表明导出的参数、分析方法正确,而微小的加速度偏差可能来源于积分算法的差异。各量的归一化均方根误差均为5%左右,完全满足工程精度要求。
由图5可知,在0~10 s阶段桥梁结构整体响应较小,阻尼器两端的相对位移较小,但阻尼器的阻尼力峰值在1000 kN左右,这是因为阻尼指数s为0.2且阻尼系数cd较大,所以即使在低速下阻尼器仍然能产生相当大的阻尼力。图5表明, Midas结果与本文二分法求解结果吻合良好,说明二分法求解能够准确获得各个时刻的阻尼力。
3.3　计算效率比较
采用MATLAB软件作为结构求解软件,利用其并行计算工具箱Parallel Computing Toolbox实现GPU代替CPU求解结构模型,验证基于MATLAB-GPU的求解性能。解算计算机的软件及硬件环境配置的详细参数见表1。
为了对比CPU计算、GPU加速计算、直接求解与显式非线性振型叠加法（ENMS法）求解数值模型的计算效率,在解算含阻尼器的桥梁模型时使用MATLAB软件的计时函数‘tic’,统计整个模型各积分步的平均耗时,结果见表2。平均耗时越短说明计算效率越高。需要说明的是,直接求解是指通过振型、节点质量、频率等信息反求出模型的刚度、阻尼矩阵,从而直接采用逐步积分算法求解结构的位移、速度和加速度响应。这样做的目的是,与ENMS法相比较,研究方法的效率。
加速比RG为CPU计算与GPU计算耗时之比,表示GPU加速效果,其数值越大表明加速效果越好。加速比RM为直接求解与ENMS法计算耗时之比,表示ENMS法提升的效率。由表2可知,采用直接求解法, GPU计算相比CPU计算可以节省大约一半时间。若采用ENMS法, GPU计算能发挥并行计算的优势,耗时显著低于CPU计算,达到约22倍的CPU计算效率。若采用CPU计算, ENMS法比直接求解计算更快,效率约为2.8倍,而采用GPU计算, ENMS法计算速度高达直接求解的32倍。同时采用2种计算方法,效率提升了约62倍。对于整体非线性不强的情况,采用显式非线性振型叠加法比直接计算效率更高,且使用GPU计算比CPU计算效率更高,二者相结合时可以大幅度缩短计算时间,提升计算效率。2种方法结合后求解效率提升巨大的原因是, GPU通常拥有数千个核心,相比CPU更适合进行多核并行计算,而通过显式振型叠加法解耦后的每个方程可独立求解,使得其符合并行计算的原则。在结构数值模型越复杂,方程数越多的情况下,能够越好地发挥GPU计算的效率优势。
3.4　不同计算规模效率比较
为了验证不同计算规模下数值模型的加速比,在MATLAB中建立串联多自由度体系模型。模型共有cn个自由度,其中m1=30 t,m2=20 t,m3~mcn为10 t。各阶振型阻尼比ζ取0.3,各个相邻质点间刚度k均相同,通过振型叠加法,由质量矩阵M和刚度矩阵K,由运动微分方程可以求解出振型矩阵Φ和各阶频率向量f。将各相邻质点间刚度k均设为108~1011 kN/m之间的某个值,使结构第1振型频率f1在2 Hz左右,进行各工况的时程计算,这一做法的目的是让该模型的固有频率更加接近大跨度桥梁,具有实际工程意义。串联多自由度体系模型的自由度分别取500自由度、1 000自由度、2 000自由度、3 000自由度、5 000自由度、7000自由度。通过MATLAB仿真对GPU加速计算与显式非线性振型叠加法展开研究,不同规模下GPU加速、ENMS法对计算效率的提升情况分别见表3、表4,不同规模下数值模型的计算耗时以及加速比对比见图6。
由表3可知,若采用直接求解,在自由度数较少的情况下, CPU计算耗时更短优势更大；自由度数达到约1000时, GPU计算耗时开始低于CPU计算；自由度数达到约2000时, GPU求解相比CPU求解可以节省约一半计算耗时。若采用ENMS法,则GPU在自由度数较小的情况下就能发挥计算优势,其耗时显著低于CPU,且其计算效率随着自由度数的上升而逐渐提高,最终可达到接近于20倍CPU计算效率。
由表4可知,无论是采用CPU计算还是GPU计算,在自由度数不多的情况下,使用ENMS法就能够发挥计算效率优势,而在使用GPU计算的情况下, ENMS法的效率优势更大,其提升的计算效率随着自由度数的上升而成倍增加,最终在7000自由度时达到接近于75倍直接计算的效率。
由图6可知,单独采用GPU加速计算或ENMS法时,计算效率的提升在不同自由度下变化较小,其加速比基本稳定在2~5左右。GPU加速计算与ENMS法结合后,结构自由度数越多,数值模型的计算规模越大,计算效率的提升越显著,其加速比甚至高达数十倍。2种方法结合后求解效率提升巨大的原因是GPU通常拥有数千个核心,相比CPU更适合进行多核并行计算,而通过显式振型叠加法解耦后的每个方程可独立求解,使得其符合并行计算的原则。在结构数值模型越复杂,方程数越多的情况下,能够越好地发挥GPU计算的效率优势,加速效果更好。
4　结论
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针对具有局部非线性的大型桥梁结构的实时计算问题,本文结合非线性振型叠加法、显式逐步积分方法、GPU加速并行计算等策略,提出了基于显式逐步积分算法与GPU加速计算的非线性振型叠加法,并开展了数值仿真研究,得出如下主要结论：
1）对于存在局部非线性的大型多自由度结构,显式非线性振型叠加法可以利用Midas Civil导出的质量与振型信息方便地进行动力响应求解,在选取合适振型的情况下,此方法的结果误差较小,精度较高。
2）对于指数阻尼的Maxwell黏滞阻尼器模型,可以使用二分法进行阻尼力的求解,能够准确求解阻尼力,较好地解决非线性阻尼器建模计算问题。
3）利用显式非线性振型叠加法对多自由度体系进行积分步解耦后,配合GPU并行计算,可显著提升数值积分计算效率。桥梁测试工况中计算效率提升62倍,且数值模型越复杂,加速计算效果越好。
4）采用显式逐步积分方法,避免了迭代求解方程；采用振型叠加法将积分步内的方程解耦,通过选取合适振型数可降低待求自由度的数量,以减少计算规模；解耦后的方程组更适合应用GPU进行加速计算,大幅度提升实时计算效率。
基金
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  • 国家自然科学基金项目(52078398)
  • 海南省重大科技计划(ZDKJ2021024)
文献
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2025年第45卷第5期
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doi: 10.13197/j.eeed.2025.0515
  • 接收时间：2024-03-23
  • 首发时间：2026-03-20
  • 出版时间：2025-10-22
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  • 收稿日期：2024-03-23
  • 修回日期：2024-05-09
基金
国家自然科学基金项目(52078398)
海南省重大科技计划(ZDKJ2021024)
作者信息
    1.武汉理工大学土木工程与建筑学院,湖北武汉430070
    2.武汉理工大学三亚科教创新园,海南三亚572000
    3.中国建筑第二工程局有限公司华中公司,湖北武汉430061

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吴斌（1970—）,男,教授,博士,主要从事结构试验方法、振动控制研究。E-mail：
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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