Uncertainty analysis of C-J detonation parameters based on polynomial chaos theory
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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。
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关键词:
- 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. -
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
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
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 
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