基于数据驱动的均匀脉冲载荷作用下低碳钢圆板动力学响应的量纲分析

doi:10.12382/bgxb.2025.0424

摘要/Abstract

摘要：

在爆炸载荷作用下,预测结构响应的关键在于准确建立输入载荷、材料性能与力学响应之间的关系。提出了一种结合数据驱动和量纲不变性分析的方法,用于识别在脉冲载荷作用下圆板动态塑性响应中的关键无量纲参数,并构建挠度预测模型。通过在ABAQUS中建立固支圆板的显式动力学模型,设置圆板半径L、厚度H、密度ρ、屈服强度σ₀及脉冲冲量I为变量生成数据;以人工神经网络拟合响应面并求梯度,结合指数矩阵法与主动子空间分析完成特征构造与降维,识别主导无量纲量。结果表明,通过指数矩阵耦合和主动子空间分析,最终将五个原始变量表示为Johnson损伤数I²/(ρσ₀H²)与几何参数H/L的组合,将问题从多维输入降维至一个核心变量。识别的无量纲量能有效表征冲击损伤和动力响应,并展现出良好的适用性。研究成果为冲击动力学问题提供了一种高效的数据驱动分析工具,体现了机器学习在工程物理问题中可能的应用前景。

关键词: 脉冲载荷, 数据驱动, 量纲分析, 动态塑性响应, 有限元模拟

Abstract:

Under explosive loading, the key to predicting a structure’s response is to accurately establish the relationship among the input load, material properties, and mechanical response. This study proposes a method that combines data-driven modeling with dimensional-invariance analysis to identify the key dimensionless parameters governing the dynamic plastic response of circular plates under impulsive loading, and to build a deflection-prediction model. An explicit dynamic model of a clamped circular plate is constructed in ABAQUS. Data are generated by varying the plate radius L, thickness H, density ρ, yield strength σ₀, and impulse I. An artificial neural network is used to fit the response surface and compute gradients; together with the exponent-matrix method and active subspace analysis, feature construction and dimensionality reduction are performed to identify the dominant dimensionless group(s). The results show that, after analysis in the principal subspace, the five original variables can ultimately be expressed as a combination of the Johnson damage number I²/(ρσ₀H²) and the geometric parameter H/L, thereby reducing a multivariable input to a single core variable. The identified dimensionless group effectively characterizes impact damage and dynamic response and exhibits good applicability. The findings provide an efficient data-driven analysis tool for impact-dynamics problems and demonstrate the potential of machine learning in engineering-physics applications.

Key words: Impulsive loading, data-driven, dimensional analysis, dynamic plastic response, finite-element simulation

葛朴昕, 宋子豪, 李志洋, 王海任, 雷建银, 刘志芳. 基于数据驱动的均匀脉冲载荷作用下低碳钢圆板动力学响应的量纲分析[J]. 兵工学报, 2025, 46(10): 250424-.

GE Puxin, SONG Zihao, LI Zhiyang, WANG Hairen, LEI Jianyin, LIU Zhifang. Data-Driven Dimensional Analysis of the Dynamic Response of Low-Carbon Steel Circular Plates under Uniform Impulsive Loading[J]. Acta Armamentarii, 2025, 46(10): 250424-.

图/表 19

表1 低碳钢材料参数

Table 1 Low-carbon steel parameters

ρ/(g·mm^-3)	E/MPa	ν	σ_o/MPa
0.007861	207×10³	0.3	233

图1 实验模拟板中心挠度对照图

Fig.1 Comparison of center deflection between experimental and simulated plates

表2 数据集参数变动范围

Table 2 Parameter variation range for the dataset

参数	符号	下限	上限
半径	L/mm	27.03	36.57
厚度	H/mm	1.64	2.22
密度	ρ/(g·mm^-3)	0.00663	0.00897
脉冲载荷	I/(N·ms)	3150	7150
屈服强度	σ₀/MPa	189.55	256.45

图2 实验与模拟数据对照图

Fig.2 Comparison of experimental and simulated data

图3 残差分布直方图

Fig.3 Histogram of residuals

图4 均匀脉冲载荷作用下低碳钢圆板动力学响应的数据驱动量纲分析示意图

Fig.4 Schematic diagram of the data-driven dimensional analysis of the dynamic response of a low-carbon steel circular plate under uniform pulse load

表3 脉冲载荷下圆板中的量纲向量

Table 3 Dimensional vector in a circular plate under pulse loading

物理量	符号	量纲向量[M,L,T]
密度/(g·mm^-3)	ρ	[1,-3,0]
初始屈服强度/(N·mm^-2)	σ₀	[1,-1,-2]
半径/mm	L	[0,1,0]
厚度/mm	H	[0,1,0]
单位面积所受脉冲/(N·ms·mm^-2)	I	[1,-1,-1]

图5 神经网络示意图

Fig.5 Schematic diagram of a neural network

表4 神经网络参数

Table 4 Neural network parameter

隐藏层	神经元数	激活函数	损失函数	优化器	批量大小
5	10	ReLU/Tanh	MAE	Adam	32

图6 训练损失和验证损失随迭代次数变化图

Fig.6 Training and validation loss versus iterations

表5 训练集、验证集与测试集性能对比

Table 5 Comparison of training, validation, and test set performance

数据集	样本数	MSE	R²	MAE
训练集	700	0.021382	0.9981	0.073878
验证集	150	0.056340	0.9952	0.092583
测试集	150	0.037933	0.9974	0.089986

图7 神经网络预测结果与实验结果对比图

Fig.7 Comparison of neural network predictions with experimental results

图8 均匀脉冲载荷下低碳钢圆盘中无量纲参数柱状图

Fig.8 Bar chart of dimensionless parameters for a low-carbon steel disk under uniform pulse loading

图9 基于SHAP的特征重要性分析结果

Fig.9 SHAP summary plot showing feature contributions

表6 基于不同特征组合的特征模型质量评估

Table 6 Quality assessment of feature models based on different feature combinations

模型	输入特征	R²	MAE/%
FE-1	($\frac{{I}^{2}}{\rho {\sigma }_{0}{H}^{2}}$,$\frac{H}{L}$)	0.97	19.7
FE-2	($\frac{{I}^{2}}{\rho {\sigma }_{0}{L}^{2}}$,$\frac{H}{L}$)	0.88	33.3

图10 低碳钢圆板在均匀脉冲载荷下的两项无量纲参数表征

Fig.10 Two dimensionless parameters characterizing a low-carbon steel circular plate subjected to a uniform impulsive load

图11 无量纲参数2与中心挠厚比W/H的拟合关系图

Fig.11 Fitting relationship between the dimensionless parameter π and the central deflection-to-thickness ratio W/H

表7 板中心挠厚比对应预测公式

Table 7 Prediction formula for the deflection-to-thickness ratio at the center of the plate

方程	$\frac{W}{H}$=a(($\frac{{I}^{2}}{\rho {\sigma }_{0}{H}^{2}}$)⁰^.²⁰⁰($\frac{H}{L}$)^-0^.³⁸²)^b
参数	a	b	R²	MSE
数值	0.558	2.361	0.9703	0.249

图12 均匀脉冲载荷下低碳钢圆盘中载荷强度I与挠厚比关系图

Fig.12 Graph of the relationship between load intensity I and the deflection-to-thickness ratio in a low-carbon steel disc under uniform impulse loading

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