爆炸冲击问题的波阵面追踪建模与数值算法

The study of explosion shock wave propagation is the key problem in explosion mechanics. Focusing on the singular problems of shock wave governing equations, the establishment of mathematical models of the wave-front tracking and numerical algorithm study through introducing arc-length parameters are performed in this project. (1) Mathematical models of explosion shock wave in arc-length space are formulated by deducing the governing equations from physical space to computational space, and then the discontinuous singularity of partial differential equations are eliminated or weakened; (2) Space transformation leads to moving meshes. Based on mesh mapping reconstruction and conservative interpolation method, the adaptive mesh of pseudo arc-length algorithm will be studied; (3) Considering the feasibility of extending to high dimension, the solving accuracy and computational efficiency in the process of simulation, it is important to select a suitable discrete method for solving the governing equations in the arc-length space, and the corresponding computational programs based on the pseudo arc-length method are developed and presented; (4)The experiments of explosion shock wave propagation and its interaction with structures are carried out in order to verify the effectiveness of our algorithm, and problems of blast wave propagation under different conditions are numerically simulated. Through comparing the quantities of both sides of wave-front, it can realize capturing and tracking the wave-front with high resolution.

爆炸冲击波传播问题的研究是爆炸力学中的核心问题。本项目针对冲击波控制方程的奇异性问题，通过引入弧长算法，开展波阵面追踪建模与数值算法研究工作：（1）开展弧长空间爆炸冲击波的数学建模研究，将物理空间的控制方程变换到弧长空间，进而消除或消弱偏微分方程的间断奇异性；（2）针对空间转化过程中导致的网格节点位置移动现象，基于网格重构映射和守恒型插值方法对伪弧长算法的网格自适应性质进行研究；（3）考虑到模拟过程中的求解精度和计算效率因素，选择适于向高维扩展的离散方法，对弧长空间控制方程进行数值求解，并开发相应的计算程序；（4）开展爆炸冲击波的传播及其与结构相互作用问题的验证性实验研究，数值模拟不同情况下的爆炸冲击波传播问题，对比分析波阵面两侧物理量的变化，实现对于波阵面的高分辨率捕捉和追踪。

本项目围绕爆炸与冲击问题中的高精度和高鲁棒性数值模拟问题开展了物理间断面追踪和数值算法的理论、数值模拟和实验研究，取得了如下的创新性成果：（1）采用伪弧长算法处理双曲型偏微分方程奇异性问题，引入弧长参数，将物理空间映射到弧长空间，提出了模拟爆炸与冲击问题的伪弧长算法；（2）针对伪弧长算法，在保持标量正值保正性和守恒性基础上，给出了相应的离散方法以及物理量向新网格的映射和重构算法，实现了网格自适应移动，精确捕捉和追踪冲击波波阵面，很好地解决了欧拉数值方法在间断面捕捉方面的问题；（3）结合网格自适应算法进行离散，开发基于伪弧长算法的爆炸与冲击问题数值模拟程序，在自适应方法保正性基础上，对爆轰问题进行了数值模拟研究；（4）开展了凝聚相炸药冲击波与结构相互作用的验证性实验，为验证伪弧长方法数值模拟程序提供了依据。