非均匀介质波动问题的人工边界处理方法研究

The analytical solutions of wave motion problems in the inhomogeneous medium are extremely hard to obtain, and only a few cases can be solved. In most situations,the problems should be solved by the numerical approaches. In the numerical analysis of wave motion problem in the inhomogeneous medium, it is very important to find some artificial boundaries with high precision and easy to apply. This project is aimed to present a series of artificial boundary techniques for the numerical simulation of wave motion problems in the inhomogeneous medium. The core works are: (1) establish the critical radiation condition for wave motion problem in the inhomogeneous medium through transforming the wave equation with variable coefficients; (2) construct the new absorbing boundary conditions which satisfy the critical radiation condition of the wave motion problem in the inhomogeneous medium, and derive the specific formulas for different numerical algorithms; (3) improve some artificial boundary conditions for less extra computation and higher practicability. The achievement in this project will help to promote the theoretical and numerical investigation of the wave motion problem in inhomogeneous medium.

获得非均匀介质波动问题的解析解极其困难，而且求解范围十分有限。大量问题需要数值求解，分析过程中非常重要又急需解决的是找到精度高且简便易用的人工边界，本课题将建立一套用于非均匀介质波动问题数值求解的人工边界处理方法。核心工作包括：(1)基于变系数波动方程给出严格的非均匀介质波动问题辐射条件；(2)构造新的满足严格的非均匀介质波动问题辐射条件的吸收边界条件，并研究其在不同数值算法环境下的具体格式；(3)改进所得人工边界条件的处理过程，减少数值求解的计算量，提高算法的实用性。本课题的研究成果将对非均匀介质波动问题的理论与数值研究起到积极的推动作用。

非均匀介质波动问题涉及到的变系数波动方程的解析求解较为困难，而非均匀介质人工边界的建立往往更依赖于数值解法。本课题首先从解析研究出发，通过变换解法确定了严格的非均匀介质波动问题辐射条件，研究了如非均匀衬砌等介质内夹杂物对弹性波的散射情况。之后基于推导出的严格的非均匀介质波动问题辐射条件构造新的吸收边界条件，自主编译程序通过有限差分法给出了吸收边界条件的具体形式，最后优化数值算法简化了计算量，提高计算效率。通过具体算例分析可以确定我们开发出来的非均匀介质人工边界条件效果较好，计算速度快，有很好的实用性。本课题的研究成果将对非均匀介质波动问题的理论与数值研究起到积极的推动作用。