可压缩多介质流的单元中心型ALE方法研究与应用

In the simulation of multi-material flows, tracing the material interfaces and dealing with the large deformation are two difficulties. In recent years the research on the ALE (arbitrary Lagrangian-Eulerian) method for these two problems becomes a hot topic. In this project, we will present a new cell-centered ALE method. This ALE method consists of three steps: a Lagrangian step, a rezoning step and a remapping step. In the Lagrangian step, the DG method is used to construct a high order Lagrangian scheme for the compressible Euler equations in Lagrangian formalism; in the rezoning step, the rezoning code will be developed based on the the rezoning algorithm presented by Knupp; in the remapping step, we will construct a high order integral conservative remapping method，then a cell-centered ALE method and the code will be obtained by coupling the above three steps for the numerical simulations of multi-material flows. This ALE method can trace the material interfaces accurately and deal with the large deformation, which provides the effective numerical method and program for the numerical simulation of the implosion dynamics, inertial confined fusion problems, and so on.

在多介质流的数值模拟中，追踪物质间界面和处理大变形问题是两大难题，近些年针对这两大难题的ALE（arbitrary Lagrangian-Eulerian）方法研究成为热点。本项目将给出一种以间断有限元法为基础的单元中心型ALE方法，该方法包含三步：拉氏步、重分步和重映步。在拉氏步，采用间断有限元法构造出拉氏框架下可压缩欧拉方程组的高精度拉氏格式；在重分步，基于Knupp提出的重分算法，研制出重分程序；在重映步，构造出一种新的高精度积分守恒重映算法，将上述三步耦合得到模拟多介质流的单元中心型ALE方法和程序。该ALE方法能够准确地追踪多介质界面和处理大变形问题，从而为内爆动力学、惯性约束巨变等问题的数值模拟提供有效的数值方法和程序。

流体力学方程组可以用来模拟内爆动力学、航空航天科学、石油工业等领域中的复杂物理问题，然而由于实际问题的复杂性，一般无法给出这些方程组的精确解，人们更倾向于采用数值方法求出流体力学方程组的数值解。本项目主要研究用有限元方法求解流体力学方程组，其内容包括研究拉氏框架下可压缩流体力学方程的间断有限元方法，带阻尼的Navier-Stokes方程的二重网格方法和稳定化两步方法，稳态Smagorinsky模型的低阶非协调有限元方法，带阻尼的瞬态Stokes方程的混合有限元方法，并通过数值算例验证了这些方法的收敛性和有效性。另外还研究了声波传输问题和弹性波传输特征值问题的数值解法。该项目为流体力学问题、声波传输问题等的数值模拟提供了新的数值方法和程序。