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## Calculation of equation of state of a material mixture

Zhou Hong-Qiang, Yu Ming, Sun Hai-Quan, He An-Min, Chen Da-Wei, Zhang Feng-Guo, Wang Pei, Shao Jian-Li
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• #### 摘要

多介质流体动力学过程的数值模拟往往涉及混合物状态方程的计算. 做图法和Newton 法是混合物状态方程计算常采用的方法, 前者虽直观精度却差, 后者计算效率高却只具有局部收敛性, 当解与其初始猜测值相差较远时Newton法不一定能够获得收敛解. 为此, 本文给出一种具有大范围收敛性的嵌入算法(imbedding method)求解混合物状态方程, 其基本思想是通过引入嵌入参数, 将待解的混合物状态方程和易解的混合物状态方程线性组合, 构成嵌入方程组, 当嵌入参数从0连续地变化到1 时, 嵌入方程组的解由易解的混合物状态方程的解连续地变化为待解的混合物状态方程的解. 嵌入方程组可由Newton法迭代求解, 也可转化为以嵌入参数为自变量的常微分方程组, 从而易于由成熟的计算方法如梯形法等进行求解. 进一步利用热力学基本关系, Maxwell形式的微分方程描述了压力和温度随嵌入参数的演化速率与应变速率和组分质量分数演化速率的关系. 对铅锡混合物热力学量的计算表明了本文算法的有效性.

#### Abstract

The problem of calculating the equation of state (EOS) of a material mixture often comes from fluid-dynamical system containing multiple materials. Generally, the EOS of a material mixture is a system of nonlinear equations which are usually solved by the tabular method and the Newton iterative method. However, the former has poor accuracy, and the later has a finite radius of convergence and hence will converge only if the initial guess is sufficiently close to the final solution. So, a procedure different from the above two method is presented for calculating the EOS of a material mixture whose constituents are in pressure equilibrium and temperature equilibrium. An imbedding method is used to determine the constituent partial thermodynamic variables subjected to the constraints that the total volume and energy of mixture and the constituent mass fractions are specified. The imbedding method has a large radius of convergence, introducing a parameter defined in the interval [0, 1] and a system of imbedding equations which is linearly composed of the to-be-solved EOS of a material mixture and the easy-to-solve EOS of a material mixture. While the parameter changes continuously from 0 to 1, the imbedding method continuously changes the solution of the to-be-solved EOS which the easy-to-solve EOS of a material mixture is continuously converted into. The system of imbedding equations can be changed into a system of ordinary differential form by taking the parameter as independent variable, easily solved by a matured computational method such as trapezoidal rule. By using thermodynamic formulae, two equations in the generalized Maxwellian form are obtained, relating respectively the pressure rate and temperature rate to the strain rate and the constituent mass fraction rate. Finally, the computational method is verified by calculating the EOS of various mass fractions of lead and tin mixture.

#### 作者及机构信息

###### 1. 北京应用物理与计算数学研究所, 北京 100094
• 基金项目: 中国工程物理研究院科学技术发展基金(批准号: 2013A0201010)和国家自然科学基金(批准号: 11272064)资助的课题.

#### Authors and contacts

###### 1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
• Funds: Project supported by the Science and Technology Development Foundation of China Academy of Engineering Physics (Grant No. 2013A0201010) and the National Natural Science Foundation of China (Grant No. 11272064).

#### 参考文献

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#### 施引文献

•  [1] Jin F Q 1999 Introduction to Experimental Equation of State (Beijing: Science Press) p1 (in Chinese) [经福谦1999实验物态方程导引 (北京: 科学出版社)] [2] Xu X S, Zhang W X 1986 Introduction to Theory of Practical Equation of State (Beijing: Science Press) p241, 423 (in Chinese) [徐锡申, 张万箱 1986 实用状态方程导论 (北京: 科学出版社)] 第241, 423页 [3] Andrews D J 1971 J. Comput. Phys. 7 310 [4] Hayes D B 1975 J. Appl. Phys. 46 3438 [5] Duvall G E, Graham R A 1977 Rev. Mod. Phys. 49 523 [6] Tang Z P 2008 Phase Transitions under Shock-Wave Loading (Beijing: Science Press) p108 (in Chinese) [唐志平 2008 冲击相变 (北京: 科学出版社)] 第108页 [7] Cox G A, Robinson C M 2009 Shock Compression of Condensed Matter-2009 in: Elert M L, Buttler W T, Furnish M D, Proud W G ed. (New York: ACP1195) p1195 [8] Kirkwood J G, Wood W W 1954 J. Chem. Phys. 22 1915 [9] Fickett W, Davis W C 1979 Detonation (Berkeley: University of California Press) p88 [10] Lee E L, Tarver C M 1980 Phys. Fluid 23 2362 [11] Johnson J N, Tang P K, Forest C A 1985 J. Appl. Phys. 57 4323 [12] Sun J S, Zhu J S 1995 Theory of Detonation Physics (Beijing: National Defense Industry Press) p403 (in Chinese) [孙锦山, 朱建士1995理论爆轰物理(北京: 国工业出版社)] 第403页 [13] Sun C W, Wei Y Z, Zhou Z K 2000 Application of Detonation (Beijing: National Defense Industry Press) p71 (in Chinese) [孙承纬, 卫玉章, 周之奎2000应用爆轰物理(北京: 国工业出版社)] 第71页 [14] Zhang B P, Zhang Q M, Huang F L 2001 Detonation Physics (Beijing: Ordnance Industry Press) p127 (in Chinese) [张宝坪, 张庆明, 黄风雷 2001 爆轰物理学(北京: 兵器工业出版社)] 第127页 [15] Stewart D S, Yoo S, Davis W C 2002 12th Symposium (International) Detonation San Diego, August 11-16, 2002 [16] Zhou H Q, Yu M, Sun H Q, Dong H F, Zhang F G 2014 Acta Phys. Sin. 63 224702 (in Chinese) [周洪强, 于明, 孙海权, 董贺飞, 张凤国 2014 物理学报 63 224702] [17] Rayleigh L 1990 Investigation of the Character of the Equilibrium of an Incompressible Heavy Fluid of Variable Density (Cambridge: Cambridge University Press) p200 [18] Chen B, Zheng Z J, Ding Y K, Li S W, Wang Y M 2001 Acta Phys. Sin. 50 711 (in Chinese) [陈波, 郑志坚, 丁永坤, 李三伟, 王耀梅 2001 物理学报 50 711] [19] Wang G Y, Zhang X H 2004 High Power Laser and Particle Beams 16 1267 (in Chinese) [王光裕, 张心宏 2004 强激光与粒子束 16 1267] [20] Wang Z Y, Li M S, Chen D Q, Xu X S 1999 Chin. J. High Press. Phys. 13 37 (in Chinese) [王正言, 李茂生, 陈栋泉, 徐锡申 1999 高压物理学报 13 37] [21] Cranfill C W 2000 LA-13661 [22] Tang G, Jiang S E, Yi Y G, Wu S C 2008 High Power Laser and Particle Beams 20 247 (in Chinese) [唐鸽, 江少恩, 易有根, 巫顺超 2008 强激光与粒子束 20 247] [23] Dahlquist G, Bjorck A Translated from the Swedish by Anderson N 1974 Numerical Methods (New York: Dover Publications INC) p252 [24] Dahlquist G, Bjorck A (Translated by Bao X S) 1990 Numerical Methods (Beijing: Higher Education Press) p297 (in Chinese) [Dahlquist G, Bjorck A (包雪松 译) 1990 数值方法, (北京: 高等教育出版社)] 第297页 [25] Zhang P W, Li T J 2007 Numerical Analysis (Beijing: Peking University Press) p128 (in Chinese) [张平文, 李铁军 2007 数值分析(北京: 北京大学出版社)] 第128页 [26] Robinson C M 2003 Shock Compression of Condensed Matter-2003 in: Furnish M D, Gupta Y M, Forbes J W ed. (New York: ACP706) p107 [27] Cox G A 2005 Shock Compression of Condensed Matter-2005 in: Furnish M D, Elert M L, Russell T P, White C T (New York: ACP845) p208 [28] Meng X J, Zong X P, Bai Y, Sun Y S, Zhang J L 2000 Acta Phys. Sin. 49 2133 (in Chinese) [孟续军, 宗晓萍, 白云, 孙永盛, 张景琳 2000 物理学报 49 2133]
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##### 出版历程
• 收稿日期:  2014-09-17
• 修回日期:  2014-10-11
• 刊出日期:  2015-03-05

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