基于多项式混沌方法对C-J爆轰参数不确定度的分析

梁霄; 王瑞利; 胡星志; 陈江涛

doi:10.11883/bzycj-2023-0030

基于多项式混沌方法对C-J爆轰参数不确定度的分析

doi: 10.11883/bzycj-2023-0030

1.
山东科技大学数学与系统科学学院，山东青岛 266590
2.
北京应用物理与计算数学研究所，北京 100094
3.
中国空气动力研究与发展中心，四川绵阳 621000

基金项目: 国家自然科学基金（12171047）；国家自然科学基金-中国工程物理研究院联合基金（U2230208）；国家数值风洞工程（NNW2019ZT7-A13）；山东省自然科学基金（ZR2021MA056）

详细信息

作者简介:
梁　霄（1984－　），男，博士，副教授，mathlx@163.com

通讯作者:
王瑞利（1964－　），男，博士，研究员， wang_ruili@iapcm.ac.cn

中图分类号: O383
计量
- 文章访问数: 275
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- PDF下载量: 54
- 被引次数: 46
出版历程
- 收稿日期: 2023-02-07
- 录用日期: 2023-07-04
- 修回日期: 2023-04-29
- 刊出日期: 2023-10-27

Uncertainty analysis of C-J detonation parameters based on polynomial chaos theory

1.
School of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, Shandong, China
2.
Beijing Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
3.
China Aerodynamics Research and Development Center, Mianyang 6221000, Sichuan, China

摘要

摘要: Chapman-Jougeut理论是预测波后爆轰物理量状态的有力工具，但以往的研究未考虑模型中的不确定因素及其影响。事实上，不确定度会影响数值模拟的预测能力和可靠性。首先，通过剖析爆轰机理，深入挖掘爆轰建模与模拟中的不确定因素。假设PBX-9502的初始密度和爆速服从对数正态分布，结合真实的试验数据，通过参数估计和Anderson-Darling假设检验法标定初始密度和爆速的概率密度函数。Beta分布用以定量刻画没有物理意义的、唯象参数的不确定度，形状参数和支集源于工程经验。Rosenblatt变换将相关的、非Gauss随机变量转化成相互独立的标准正态分布。然后，使用非嵌入多项式混沌研究高维爆轰不确定度传播。具体而言，针对一元多项式混沌，正交多项式通过Gauss-Hilbert空间中的Gram-Schmidt方法导出，六点Gauss求积方法用以计算多项式混沌的系数。使用权重和Gauss求积点的全张量积计算多元多项式混沌。最后，通过多元多项式混沌得到感兴趣量的概率密度函数以及对应的期望、标准差和置信区间等Gauss统计量。研究结果表明：波后压力波动较大，置信区间较宽，与孙承纬的“爆轰压力测量值分散性较大”的结论相吻合。同时感兴趣量的试验结果落入模拟结果的置信区间内，研究结果能增强模型的可靠性和鲁棒性。所用方法可扩展到更加复杂状态方程的爆轰系统。
- 非嵌入多项式混沌 /
- 不确定度量化 /
- Rosenblatt变换 /
- Anderson-Darling检验 /
- Chapman-Jougeut理论
Abstract: The Chapman-Jouguet theory is a powerful tool to predict the states of physical quantities at the rear of the shock front. However, uncertain factors and their influences on the system response quantities are neglected in the model of previous studies. Actually, the reliability and predictability of numerical simulation will be greatly affected by these uncertainties. To begin with, uncertainties of modeling and simulation of detonation process is discussed based on the detonation mechanism. Initial density and detonation velocity of PBX-9502 are assumed to satisfy the logarithmic normal distribution. The probability density functions (PDFs) of initial density and detonation velocity are derived from Anderson-Darling hypothesis test and parameter estimation combined with real experimental data. Beta distribution is utilized to cope with empirical parameters which have no physical meaning at all, with shaping parameters and supporting set are given according to the engineer’s experience. Rosenblatt transformation is used to transform the dependent and non-Gaussian random variables into independent standard Gaussian random variables. Furthermore, nonintrusive polynomial chaos (PC) method is used to study high dimensional uncertainty propagation of detonation waves. In particular, as for one variable PC, orthogonal polynomials are derived through Gram-Schmidt algorithm in Gauss-Hilbert space, Gauss integral formula with six quadrature points is used to compute coefficients of PC. Full tensor product of quadratures and weights is applied in PC of multivariate. PDF and corresponding Gaussian statistics such as expectation, standard deviation and confidence interval of quantity of interest (QoI) are obtained from the multivariate polynomial chaos. The result shows that the variation of detonation pressure is larger and the range of confidential interval is wider. It coincides with Professor Chengwei Sun’s conclusion that “The discreteness of detonation pressure is larger in experimental measurement”. The experimental data falls into the confidential interval of QoIs, then the reliability and robustness of the modeling is enhanced. And the methodology can be extended to the detonation system with much more complex equation of state.
- non-intrusive polynomial chaos /
- uncertainty quantification /
- Rosenblatt transformation /
- Anderson-Darling test /
- Chapman-Jouguet theory

HTML全文

图 1 爆轰波在高能炸药中的传播示意图

Figure 1. Sketch of propagation of detonation wave into high explosive

下载: 全尺寸图片幻灯片

图 2 PBX-9502初始密度的概率密度函数

Figure 2. PDF of initial density of PBX-9502

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图 3 PBX-9502爆速的概率密度函数

Figure 3. PDF of detonation velocity of PBX-9502

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图 4 多方指数γ的概率密度函数

Figure 4. PDF of polytrophic exponent γ

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图 5 不确定分析算法流程图

Figure 5. Flow chart of uncertainty analysis

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图 6 二维全张量积Gauss-Hermite求积点的位置

Figure 6. Locations of Gauss-Hermite quadrature points used for two-dimensional tensor product

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图 7 前6阶单变量Hermite多项式

Figure 7. The first six orders of univariate Hermite polynomials

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图 8 系统响应量的概率密度函数

Figure 8. PDF of system response quantities

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表 1 6节点Hermite-Gauss积分的求积点和权重^[21]

Table 1. Quadrature points and weights for Hermite-Gauss integration with six points^[21]

$\xi_{r}$	$w_{i}$	$\xi_{r}$	$w_{i}$	$\xi_{r}$	$w_{i}$
–3.324 26	0.002 56	–0.616 71	0.408 83	1.889 18	0.088 62
–1.889 18	0.088 62	0.616 71	0.408 83	3.324 26	0.002 56

下载: 导出CSV

表 2 波后感兴趣量的试验和统计结果

Table 2. Statistical and experimental result of QoIs at the rear of the shock wave

波后感兴趣量	ρ_J/(g·cm⁻³)	U_J/(m·s⁻¹)	p_J/GPa
平均值	2.529	1 943	28.410
置信区间下限	2.524	1 789	24.010
置信区间上限	2.534	2 108	33.390
试验数据^[24]	2.525	1 922	28.301
标准差	0.003	81	2.376

下载: 导出CSV

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施引文献

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