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Article(id=1241838550027669596, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.055, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1730736000000, receivedDateStr=2024-11-05, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774006982571, onlineDateStr=2026-03-20, pubDate=1745337600000, pubDateStr=2025-04-23, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774006982571, onlineIssueDateStr=2026-03-20, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774006982571, creator=13701087609, updateTime=1774006982571, updator=13701087609, issue=Issue{id=1241838547544641612, tenantId=1146029695717560320, journalId=1241755870837649424, year='2025', volume='46', issue='2', pageStart='149', pageEnd='296', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774006981978, creator=13701087609, updateTime=1774007061443, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241838880882758461, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241838880882758462, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1241838547544641612, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=275, endPage=285, ext={EN=ArticleExt(id=1241838550434517089, articleId=1241838550027669596, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Distance-Minimizing Data-Driven Method for Solving Structural Dynamic Response, columnId=1241831201674171363, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Papers, runingTitle=null, highlight=null, articleAbstract=

The distance-minimizing data-driven method introduces a new computing paradigm and focuses on computational solid mechanics research. This method enables direct input of discrete material data sets (stress-strain pairs), bypassing the empirical constitutive modeling process and reducing modeling errors and uncertainties. To apply this method to boundary value problems, it is necessary to define a distance functional from the solution set to the material data set, seeking the functional extremum that satisfies the strain-displacement relationship and equilibrium equation from the material data set. In this study, we extend the method to structural dynamics, using the structural dynamic equilibrium equation as a constraint for the distance functional. We derive data-driven computing formulas, analyze the value of the constant matrix in the formula, and develop an algorithm for solving structural dynamic responses. The accuracy and efficiency of the proposed method are validated through linear and nonlinear dynamic response analyses of single-degree-of-freedom systems and multi-degree-of-freedom truss structures. Within this theory, the final value from the previous moment serves as the initial value for the current moment, facilitating faster numerical solutions and reducing computational time. Additionally, the material data set accommodates both linear and nonlinear material behaviors. It is also found that when the amount of material data exceeds 100, the amount of material data and excitation step minimally impact computational accuracy, with the signal-noise ratio (SNR) becoming the primary factor. Under the same conditions, the amount of material data and excitation step significantly affect computational efficiency, while the influence of SNR can be ignored. This study provides theoretical support for the development of data-driven dynamic solvers.

, correspAuthors=Dongfa Sheng, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Biao Dong, Guolin Xu, Dongfa Sheng), CN=ArticleExt(id=1241838572756603566, articleId=1241838550027669596, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=求解结构动力响应的距离最小化数据驱动法, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

距离最小化数据驱动法作为计算力学的新范式，在预测结构响应方面优势显著. 该方法直接输入离散的材料数据集（一组应力-应变对），绕开了传统经验本构建模，通过定义解集到材料数据集的距离泛函，并从材料数据集中搜索满足应变-位移关系和平衡方程的泛函极值，实现边值问题求解. 距离最小化数据驱动法在无阻尼振动分析中已取得成功，但还不能用于求解有阻尼振动问题. 本研究采用含阻尼项的结构动力平衡方程作为距离泛函的约束条件，使用上时刻的终值作为当前时刻的初值代替随机初值方案，提出一种求解结构动力响应的距离最小化数据驱动法. 以单自由度体系和多自由度桁架为例，通过线性和非线性弹性材料动力响应分析，验证了本文方法的准确性和高效性.

, correspAuthors=盛冬发, authorNote=null, correspAuthorsNote=
**E-mail：.
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Orthogonal experimental table of three factors and three levels

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FactorsTest 1Test 2Test 3Test 4Test 5Test 6Test 7Test 8Test 9
Δt0.020.020.020.010.010.010.0050.0050.005
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R1%5%3%5%3%1%3%1%5%
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三因素三水平的正交试验表

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R1%5%3%5%3%1%3%1%5%
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求解结构动力响应的距离最小化数据驱动法
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董标 , 徐国林 , 盛冬发 **
固体力学学报 | 研究论文 2025,46(2): 275-285
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固体力学学报 | 研究论文 2025, 46(2): 275-285
求解结构动力响应的距离最小化数据驱动法
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董标, 徐国林, 盛冬发**
作者信息
  • 西南林业大学土木工程学院，昆明，650224

通讯作者:

Distance-Minimizing Data-Driven Method for Solving Structural Dynamic Response
Biao Dong, Guolin Xu, Dongfa Sheng**
Affiliations
  • Institute of Civil Engineering, Southwest Forestry University, Kunming, 650224
出版时间: 2025-04-23 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.055
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摘要
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距离最小化数据驱动法作为计算力学的新范式，在预测结构响应方面优势显著. 该方法直接输入离散的材料数据集（一组应力-应变对），绕开了传统经验本构建模，通过定义解集到材料数据集的距离泛函，并从材料数据集中搜索满足应变-位移关系和平衡方程的泛函极值，实现边值问题求解. 距离最小化数据驱动法在无阻尼振动分析中已取得成功，但还不能用于求解有阻尼振动问题. 本研究采用含阻尼项的结构动力平衡方程作为距离泛函的约束条件，使用上时刻的终值作为当前时刻的初值代替随机初值方案，提出一种求解结构动力响应的距离最小化数据驱动法. 以单自由度体系和多自由度桁架为例，通过线性和非线性弹性材料动力响应分析，验证了本文方法的准确性和高效性.

数据驱动计算力学  /  非线性动力学  /  力学变分法  /  泛函极值  /  离散本构

The distance-minimizing data-driven method introduces a new computing paradigm and focuses on computational solid mechanics research. This method enables direct input of discrete material data sets (stress-strain pairs), bypassing the empirical constitutive modeling process and reducing modeling errors and uncertainties. To apply this method to boundary value problems, it is necessary to define a distance functional from the solution set to the material data set, seeking the functional extremum that satisfies the strain-displacement relationship and equilibrium equation from the material data set. In this study, we extend the method to structural dynamics, using the structural dynamic equilibrium equation as a constraint for the distance functional. We derive data-driven computing formulas, analyze the value of the constant matrix in the formula, and develop an algorithm for solving structural dynamic responses. The accuracy and efficiency of the proposed method are validated through linear and nonlinear dynamic response analyses of single-degree-of-freedom systems and multi-degree-of-freedom truss structures. Within this theory, the final value from the previous moment serves as the initial value for the current moment, facilitating faster numerical solutions and reducing computational time. Additionally, the material data set accommodates both linear and nonlinear material behaviors. It is also found that when the amount of material data exceeds 100, the amount of material data and excitation step minimally impact computational accuracy, with the signal-noise ratio (SNR) becoming the primary factor. Under the same conditions, the amount of material data and excitation step significantly affect computational efficiency, while the influence of SNR can be ignored. This study provides theoretical support for the development of data-driven dynamic solvers.

data-driven computational mechanics  /  nonlinear dynamics  /  mechanical variation  /  functional extremum  /  discrete constitutive model
董标, 徐国林, 盛冬发. 求解结构动力响应的距离最小化数据驱动法. 固体力学学报, 2025 , 46 (2) : 275 -285 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.055
Biao Dong, Guolin Xu, Dongfa Sheng. Distance-Minimizing Data-Driven Method for Solving Structural Dynamic Response[J]. Chinese Journal of Solid Mechanics, 2025 , 46 (2) : 275 -285 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.055
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0　引言
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近年来，数据科学和机器学习的快速发展促进了计算力学的进步，为构建便捷高效的计算力学模型开辟了广阔前景[1,2]. 其中，将数据科学与计算力学相结合的数据驱动计算力学引起国内外学者的普遍关注. 数据驱动计算力学[3,4]的概念最早于2016年由Kirchdoerfer和Ortiz提出，这一概念被认为是继实验力学、理论力学和计算力学之后的第四范式[5]. 它绕开了经典计算力学中材料本构建模和模型选择的过程，以测量的材料数据作为离散本构，消除了经验本构建模的主观性和不确定性.
数据驱动计算力学中的距离最小化数据驱动法是用测量的材料数据（应力-应变）表征材料力学行为，并从材料数据中搜索泛函极值作为单元内待求的应力和应变. 该方法的计算精度和计算效率分别受到材料数据集和搜索算法的影响. Leygue等[6,7]提出数据驱动识别法，运用数字图像相关技术[8]识别试件表面的全场位移，并将位移和外载荷输入到数据驱动计算框架获得高精度的材料数据. Tang等[9]提出MAP123方法，通过分解一维材料数据获得能够描述三维空间力学行为的多维材料数据. 然而，这些材料数据中的噪点或异常点可能引起计算结果出现较大误差甚至失真. 针对这一问题，Kirchdoerfer和Ortiz等[10]引入最大熵原理对材料数据进行降噪处理，提高求解算法的鲁棒性. 在提升搜索效率方面，主要是优化数据结构和改进搜索算法. 阳杰[11]和Korzeniowski等[12]将材料数据集分割为多个子数据集，应用分层搜索算法提高计算效率；Yanchuan等[13]基于CUF（Carrera Unified Formula）理论建立新的数值模型以减少高斯积分点和插值点，并结合树搜索算法提升计算效率；Eggersmann等[14]对比分析了k-d树、k-means树和k-NN算法对计算效率的影响，指出当材料数据满足最大熵特征时，k-means树是最适合的搜索方案；Nguyen等[15]从材料数据中抽象出隐式本构流形，并用其代替原始材料数据以降低计算成本；Bessa等[16]提供了一种将多维材料数据映射到低维空间的聚类算法，用来提升多维力学行为分析的计算效率. 目前，距离最小化数据驱动法已推广应用到多个方面，如有限弹性分析[17-19]、非线性弹性分析[20,21]、非弹性分析[22]、多尺度分析[23]、塑性分析[24]、脆性断裂问题[25-27]以及突跳问题[28]等，此外，在开孔泡沫结构[29]、编织物增强复合材[30]、岩石材料[31]和细胞材料[32]等方面也得到应用.
现有研究侧重于静力学领域，以静力平衡方程作为距离泛函约束条件，忽略了加速度和速度影响. 然而，结构动力响应分析中，惯性力是重要的影响因素，不能忽略. Ortiz教授[33]及其团队通过在静力平衡方程上加入惯性力，将静力学的距离最小化数据驱动法推广到动力学领域，成功求解无阻尼振动问题. 但此方法没有考虑振动衰减特性，不能用于求解有阻尼结构振动问题.
本文采用结构动力平衡方程作为距离泛函的约束条件，推导泛函变分，尝试提出一种求解有阻尼振动问题的距离最小化数据驱动法. 将通过单自由度线性动力响应分析和桁架结构非线性动力响应分析，验证所提方法的有效性.
1　理论公式
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本节，先回顾无阻尼的距离最小化数据驱动法，再引入含阻尼项的结构动力平衡方程，推导适用于有阻尼振动分析的数据驱动计算公式，最后讨论计算公式中量纲平衡常数矩阵的取值.
1.1　无阻尼振动分析的距离最小化数据驱动法
Kirchdoerfer和Ortiz教授提出用于求解无阻尼振动问题的距离最小化数据驱动法[33]. 该方法是将边值问题转变为泛函极值问题，以动力平衡方程和应变-位移关系作为泛函的约束条件，在预先给定的材料数据集中寻找结构最匹配的应力-应变状态. 全局距离泛函可以定义为
式中，εe，k和σe，k分别表示单元e在第k时刻的应变矢量和应力矢量，和为材料数据集中的应变矢量和应力矢量，为单元e的第i个积分点权重，为量纲平衡常数矩阵，g为积分点数. 上标‘T’和‘-1’分别表示矩阵转置和取逆. 单元应变-位移关系和动力平衡方程分别为
式中，Be为单元应变矩阵，Fk为第k时刻的外力矢量，M为整体质量矩阵，ue，k为单元节点位移，为整体节点加速度，m为单元数. 下标‘e’表示单元矢量，没有下标‘e’表示整体矢量. ‘Σ’表示矩阵组装运算，例如，单元节点等效力通过‘Σ’运算可以得到整体节点等效力. 式（3）的等式左边表示单元应力矢量计算的整体等效节点力.
式（2）带入式（1）得出
此时，若能够从材料数据集中找到一组使式（4）取得极值，即可实现边值问题求解. 然而，式（4）作为多元函数，需要引入含变量ue，k和σe，k的约束条件才能求解. 因此，对式（3）应用Newmark法获得约束条件. 若第k时刻的加速度和第k-1时刻的位移uk-1、速度和加速度已知，则由Newmark法预测第k时刻的位移uk和速度为
式中，β和γ为积分参数. 当未知时，由Newmark法预测第k时刻的位移和速度为
联立式（5）和（6），可得
把（6a）式代入式（3），消去得到动力平衡方程的代数方程
以式（6）作为式（4）的约束条件，引入拉格朗日乘子数组ηk，建立拉格朗日函数
为了求解式（9），把ue，k、σe，k、ηk作为独立变量进行变分. 当Π达到驻值时（δΠ=0），可以导出
把式（10a）带入式（10c），消去待定量σe，k可得
联立式（10b）和（11），建立只包含uk和ηk的方程组
求解式（12）可获得节点响应位移uk和拉格朗日乘子ηk，再将其带入到式（2）和（10a）求出应变和应力，即可完成动力响应计算.
1.2　有阻尼振动分析的距离最小化数据驱动法
当考虑阻尼作用时，结构动力平衡方程为
式中，C为阻尼矩阵. 把式（7）带入式（13），将微分方程转变为代数方程
以式（14）作为（4）式的约束条件，建立拉格朗日函数
当Π达到驻值时（δΠ=0），可以导出
把式（16a）带入式（16c）得出
整理式（16b）和（17）可得
结合式（18）、（2）和（16a）可求出结构动力响应.
1.3　量纲平衡常数矩阵CC的取值
量纲平衡常数矩阵用于度量当前位形下两个应力应变状态之间的距离远近. 在卡氏坐标系中，空间内任意两点之间距离的计算方法众所周知. 若对微元体内两个应力应变状态（ε，σ）和（ε*，σ*）直接应用空间内两点距离的计算方法，其计算结果并不能代表这两个状态的距离. 这是因为卡氏坐标系中坐标轴的量纲均为单位1，而应力应变的量纲并不相同. 因此，引入量纲平衡常数矩阵实现量纲统一. 设和为两个独立常数，用S代替. 确定和S，一般有如下三种思路：
（1）取S=1，则的量纲为M2 L-2 T-4，此时可取，其中E为弹性矩阵，量纲为ML-1 T-2.
（2）取，则S的量纲为T4 L2 M-2，此时可取S=D2，其中D为柔度矩阵，D=E-1.
（3）将应变项和应力项均转化为能量密度，其中应变能密度和余能密度的量纲均为ML-1 T-2. 此时的量纲为ML-1 T-2，S的量纲为T2 LM-1. 建议取，S=D. 这种取值方法可使与S满足互逆关系.
以上3种平衡常数的取值方案中，使用前两种方案计算式（12）和（18）中含平衡常数项所得物理量没有实际意义，而使用第三种方案得到的物理量可表示刚度矩阵和等效节点力. 本文选用第三种取值方案. 需要注意，距离罚函数中应变能密度和余能密度不代表物体实际的能量密度，仅用于衡量两个应变应力状态之间的距离.
2　求解程序
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在试验的材料数据集中取出一组作为所有单元的应力应变，对比同时满足应变-位移关系和平衡方程的应力应变状态（εe，σe），若所有单元的到（εe，σe）之间的距离均最小，则认为是物体内待求的应力应变. 根据上述推导的计算公式，编写数据驱动求解器. 数据驱动求解器的算法流程如图1所示，具体计算步骤如下：
（1）输入质量矩阵M、阻尼矩阵C、刚度矩阵K、外力矢量F和材料数据集，设定激励步长Δt、分析时长、位移初值u0、速度初值、加速度初值、Newmark参数β、γ；
（2）选取一组作为所有单元的应力应变初值；
（3）由式（6）计算位移和速度；
（4）根据计算等效节点力；
（5）由式（18）计算位移uk和拉格朗日乘子ηk；
（6）根据式（2）和（16a）计算（εe，σe），并在材料数据集中搜索距离（εe，σe）最近的数据点，判断与是否相等：
（6.1）否，令，回到步骤（3）继续迭代；
（6.2）是，结束循环；
（7）回到步骤（2）继续迭代，直至计算出所有时刻的结构响应.
（8）输出节点位移uk、单元应变应力.
任意时刻开始计算前，都需要设置应力应变初值. 现有的随机初值方案从侧面说明了初值选取不会改变最终的计算结果. 鉴于动力分析中相邻时刻材料的应力状态和应变状态具有邻近性，本文改进初值选取方案：第一时刻采用零值作为初值，而其余时刻采用上时刻的终值作为初值. 从理论角度分析，采用改进方案选取的应力应变初值更接近最终解，因此所需迭代搜索次数相较于随机方案理应更少. 这一点将在后续分析中得到验证.
3　算例验证
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本节，首先计算线弹性单自由度体系和非线性弹性桁架的动力响应，验证本文方法的计算精度；然后，分析材料数据量、激励步长和信噪比对结果精度和计算成本的影响；最后，比较随机初值方案和本文初值方案在程序执行过程中所需的计算时间，说明本文方法在计算时间上的优势. 这里，信噪比表示叠加的随机噪声与拟合本构在同一应变上应力大小的比值，用来模拟测试应力-应变点的分布情况.
3.1　线弹性单自由度体系
考虑悬臂立柱体系（图2（a））中质点在水平自由度上的振动情况，可将其简化为弹簧-阻尼-质点体系（图2（b））. 假设质点只有水平自由度，可使用拉压杆单元等效弹簧（图2（c））. 拉压杆单元右端处的水平自由度上，刚度矩阵的分量为k，质量矩阵的分量为m. 设拉压杆单元的弹性模量为2×108 N/m2，截面面积为8×10-4 m2，密度为3×106 kg/m3. 以线弹性本构关系生成离散的无噪声应力应变数据，代替线性本构关系表征材料力学行为. 在质点的水平方向上施加动态荷载，荷载大小为f（t）=3×104 e-100（t-0.5）2 N. 根据单元属性计算出质点的等效质量、等效刚度和等效阻尼，分别为1200 kg、1.6×105 N/m、3500.
使用精细积分法（PIM）、Newmark法（取β=0.25，γ=0.5）、Wilson-θ法（取θ=1.4）以及本文的距离最小化数据驱动法（Data-driven method，DDM）计算单自由度体系的动力响应，响应位移时程如图3所示. 从图3可以看出，DDM曲线和其它三条曲线基本重合. 这表明本文方法与现有方法具有相同的计算精度.
3.2　非线性弹性桁架
计算非线性弹性桁架结构的动力响应，分析本文方法的计算精度. 由于距离最小化数据驱动法无条件收敛，其计算结果是否满足精度要求与材料数据量大小有关，学者们普遍认为材料数据量低于100时计算精度难以满足要求[19]，因此本文研究选用的材料数据量不低于100. 桁架结构如图4所示，其跨度l为6 m，高h为1 m，节间长度d为1 m. 跨中下节点受动荷载作用，荷载时程为f（t）=3×104 e-100（t-0.5）2 N. 桁架单元的密度为3×106 kg/m3，弹性模量为2×1010 N/m2，泊松比为0.3，截面面积为8×10-4 m2. 采用瑞利阻尼控制振动衰减，阻尼参数为a0=0.25，a1=0.01. 选取积分参数β=0.25、γ=0.5，激励步长为0.005 s，计算时长为5 s，位移、速度和加速度的初值为零. 假设材料的拉伸和压缩曲线中心对称，根据超弹性Mooney-Rivlin模型生成均匀分布的应力-应变数据，并将其与随机噪声叠加，获得测试材料数据集，具体参数如图5所示. 选用的随机噪声强度范围设定为最大应力的0~5%之间. 输入测试材料数据集，使用本文方法（DDM）计算桁架结构的动力响应. 对比基于超弹性Mooney-Rivlin模型使用有限元软件（FEM）计算获得的动力响应，并绘制两种方法在跨中下节点y方向上的响应位移时程于图6.
由图6可知，DDM曲线和FEM曲线均随时间呈振荡衰减趋势，但幅值和相位的差异均随着分析时长增加而增大，在峰值处尤为明显. 作者认为这种差异由以下三个方面因素引起：一是，材料数据集的精细度不足，不能精确描述材料力学行为；二是，激励步长影响误差积累量；三是，噪声引起的计算结果误差.
3.3　桁架响应的精度分析
使用3.2节提到的桁架结构，分析材料数据量（N）、激励步长（Δt）和信噪比（R）对计算精度影响. 同3.2节一样，计算精度分析在材料数据量不低于100的情况下进行. 将试验分组参照三因素三水平的正交试验表（表1），使用本文方法计算各试验分组下结构的动力响应位移时程（图7），并以有限元方法计算的动力响应位移作为对照组（Δt=0.005 s），如图7中的灰色实线.
由图7可知，在材料数据量不低于100的情况下，信噪比是影响本文方法计算精度的主要因素，而材料数据量和激励步长的影响相对较小；信噪比为1%的三条曲线相较于其它曲线表现出更高的计算精度，但是随着信噪比的增大，计算精度逐渐降低；此外，随着分析时长增加，所有试验分组的结果精度均有所下降. 综上分析，材料数据中的噪声或异常点会对结构动力响应精度产生显著影响，特别是在中长周期内，误差累积问题会进一步放大这种影响. 因此，计算分析前，建议对实测的材料数据进行降噪处理.
3.4　桁架响应的计算成本分析
在3.3节基础上，统计各试验组在整个分析过程中的迭代次数频数分布和计算时间，如图8所示. 由图8可知，所有试验组迭代次数的频数分布均较为集中，在5~11次之间；相同激励步长下，数据量每增加一个量级，任意时刻的计算次数增加约1~2次，总计算时间增加0.25~0.83倍，总迭代次数增加0.03~0.38倍，其中，0.03倍发生在编号8和9之间；相同数据量下，激励步长每缩小一半，任意时刻的迭代次数基本相同，总计算时间增加约0.9~1.3倍，总迭代次数增加约0.7~1.3倍. 这说明材料数据量大小和激励步长是影响计算效率的主要因素，而噪声影响较小可以忽略.
3.5　两种初值设定方案的计算时间对比
进行对比的两种初值设定方案分别是改进方案和随机方案. 改进方案是在第一时刻使用零值作为初值，其余时刻采用上一时刻的终值作为初值. 随机方案在任意时刻的初值都从材料数据集中随机选择. 图9展示了改进方案（Optimize）和随机方案（Random）的计算时间成本. 从图9可以明显看出，采用改进方案所需的计算时间大幅减少，减少幅度在35%到63%之间. 此外，随着激励步长的减小和材料数据量的增大，计算时间的减少幅度更为显著.
4　讨论与结论
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基于经验本构模型计算结构动力响应的有限元方法是结构抗震分析与设计的重要组成部分，但经验本构模型在建模和参数选取时，会引入主观误差和不确定性，进而影响结构动力响应的计算精度. 基于深度学习的数据驱动法，通过训练神经网络可以获得材料参数与力学响应的映射关系，此方法需要消耗大量计算资源. 本文研究的距离最小化数据驱动法可以直接输入离散的材料数据集（一组应力-应变对）进行计算，绕开了经验本构建模和训练神经网络等工作，以新的求解思路避开了前两种方法存在的问题. 使用材料数据集表征材料力学行为，不再局限于特定的函数形式，适用于描述极端环境（高温、高压、高速等）下材料内部宏观、微观力学行为. 该方法是从物质的基本准则和守恒定律中推导而来，这为计算结果的准确性和可解释性提供了坚实的理论基础. 距离最小化数据驱动法在求解结构响应方面显现计算优势，但当前研究忽略了结构振动的衰减特性，不能用于求解有阻尼结构振动问题.
本文采用含阻尼项的结构动力平衡方程，改进距离最小化数据驱动法以求解有阻尼振动问题. 通过线弹性单自由度体系和非线性弹性桁架的动力响应计算分析，验证了本文方法的准确性. 同时发现，当材料数据量超过100时，材料数据量和激励步长对计算精度影响小，信噪比成为影响计算精度的主要因素. 相同条件下，材料数据量和激励步长对计算效率影响大，而信噪比影响则可以忽略. 值得注意的是，本文使用的数值算例相对简单，没有采取抑制噪声的有效措施，也没有考虑非弹性和路径依赖的材料行为. 后续工作将采用更复杂的分析算例，增强算法对噪声或异常点的鲁棒性，考虑非弹性和路径依赖的材料行为，进一步验证本文方法的准确性.
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  • 云南省教育厅研究生基金项目(2023Y0777)
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2025年第46卷第2期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.055
  • 接收时间：2024-11-05
  • 首发时间：2026-03-20
  • 出版时间：2025-04-23
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  • 收稿日期：2024-11-05
基金
云南省教育厅研究生基金项目(2023Y0777)
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    西南林业大学土木工程学院，昆明，650224

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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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