基于特征理论的固体炸药爆轰单元中心型拉格朗日计算方法研究

To improve the fidelity of the numerical simulation for two-dimensional detonation flows in solid explosive, a new cell-centered Lagrangian hydrodynamic method with spatial-temporal second-order precision is proposed. The main feature of this numerical method is that the velocity at the cell vertex results from the local evolution solutions computed by the characteristic theory about two-dimensional partially differential equations for the explosive detonation flows, and the corresponding finite volume scheme has “genuinely multidimensional nature” of a general hyperbolic system. During the local evolution solutions, the exact integral equations for the linearized hyperbolic system were derived from the general theory of characteristics in terms of the primitive physical variables, and these integral equations can be solved approximately, and then the simplified analytic expressions be derived by means of a limit operation about evolving for an infinitely small time interval. This nodal solver from the characteristic theory essentially is a multidimensional Riemann solver or a generalization of the original idea of Godunov to multidimensional hyperbolic conservation laws. The spatial discretization with second-order precision is achieved by means of the choice about the calculating technique of the bilinear interpolation gradients, the construction about the gradient limiters of the scalar variables and vector variables, and the design about evaluating way of the numerical fluxes across grid edge, and it can preserve the strict monotonicity and fine robustness and satisfy the entropy inequality. The temporal discretization with second-order precision is achieved by means of the choice about implicit-explicit multistage Runge-Kutta scheme to solve the ordinary differential equations with strongly stiff source term, and it possesses a strong stability preserving (SSP) nature. And then, the positivity-preserving property of the second-order discretized scheme is obtained by imposing a positivity-preserving limiter, in particular, the mass fraction of chemical reaction can also preserve positivity. From some representative test cases, the new cell-centered Lagrangian method is demonstrated its precision, high resolution, convergence, stability, positivity, conservation, robustness, and so on. The presented method is able to advance the level and capability of the numerical simulation for detonation flows in solid explosive, and further provide the strong support for the design, research, assessment and authentication of nuclear weapon devices.

针对二维固体炸药精密爆轰物理亟待提高数值模拟置信度这一应用背景，本项目提出一个新型的时空二阶精度的单元中心型拉格朗日计算方法，该方法主要特点是：网格节点速度由二维爆轰反应流动方程特征理论计算出的局域演化解确定，由此给出的有限体积法具有“真正多维性质”。在此基础上，通过变量插值梯度计算方案的选用、具有“真正多维性质”的标量梯度限制器与矢量梯度限制器的构造以及数值通量求解方案的设计，给出了单调性及网格健壮性均良好并满足熵增条件的二阶空间离散格式；再通过带强刚性源项常微分方程系统显-隐Runge-Kutta法的选用，给出了强稳定性保持的二阶时间离散格式；最后通过对二阶显-隐Runge-Kutta法获得的离散表达式进行保正限制器的重构处理，确立了数值格式具有保正性质。新型单元中心型拉格朗日算法能够提高固体炸药爆轰数值模拟水平与能力，也可进一步为核武器装置的设计、研制、评估、认证等研究工作提供支持。

固体炸药爆轰在国防科技和国民经济，特别在核武器研究中发挥极其重要作用。核禁试以来，美国等西方发达国家实施了诸如“先进模拟与计算”的庞大科技计划，力图通过更精密的实验室模拟实验和更精确的大规模科学计算来维护和发展其核武库，其中，更高置信度的固体炸药爆轰数值模拟方法是非常重要的一个研究领域。对我国而言，由于实验技术水平和科学计算能力的整体落后，发展更高置信度的固体炸药爆轰数值模拟方法显得更加迫切，高置信度数值模拟有助于对核武器装置的设计、研制、评估、认证等研究工作提供强有力的保障和支持。.直接根据二维固体炸药爆轰反应流动方程组的特征理论来计算网格节点运动速度是一种非常理想的数值模拟方式。本文构造的计算格式主要思路是：在中心型拉格朗日系统及有限体积方法的框架下离散爆轰反应流动方程组，根据爆轰方程的特征性质求解离散网格节点的速度与压力，获得的网格节点物理量被用来更新网格节点位置以及计算网格单元边的数值通量。利用特征理论得到的网格节点解形式简单，同时真正考虑了多维效应，是一维 Godunov 格式在多维问题中的推广。文在模拟凝聚炸药的爆轰过程中使用了点火成长爆轰模型，化学反应速率选用三项点火成长方程，固体炸药和气体爆轰产物使用 JWL 状态方程。获得的计算方法具有以下优势：1）单元中心型拉式方法应用了守恒律，在计算过程中能够保持系统动量与总能量的守恒性，并且不需要人工粘性即可捕捉爆轰波；2）半隐式龙格库塔迭代具有保正性和强稳定性，结合分片线性重构，使格式达到了高阶精度，保证良好收敛性的同时具有较高分辨率；3）根据爆轰方程特征性质构造的节点解法器，形式简单计算量小，并且能够真正体现爆轰流动双曲型偏微分方程的“多维特性”；4）计算方法对网格没有限制，可以在结构网格或非结构网格中使用，灵活性好应用范围广；5）计算过程中各求解步骤互相独立没有耦合迭代，易于模块化编程优化及应用大规模并行计算。.典型算例的计算结果表明，本文构造用于模拟凝聚炸药爆轰的基于特征理论的高精度单元中心型拉式方法具有较高的的的计算精度与很好的守恒性，能够清晰准确地刻画爆轰波的传播过程以及化学反应区内物质的流动规律，也能够准确细致地反应二维爆轰波相互作用的物理过程。

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