摘要
半整数维波动方程体现的是一种由点及线、由线及面、由面及体的抽象思维方式,能够在较小求解成本、较高求解效率下揭示各种形式的波在空间中的传播特性,然而其内在机理相对复杂、对细节把控要求较高,其求解具有一定的技术门槛。以频域有限元方法为例,从1.5维声波、弹性波方程及其耦合形式的推导起步,对完全匹配层(PML)拉伸尺度选取、等效积分弱形式推导、耦合边界条件设置、极点规避等技术细节进行注记。采用声波与弹性波耦合的1.5维波动方程对二维问题进行求解,频域有限元数值解与解析解高度吻合。思想方法容易延用到由2.5维波动方程表征的,基于其他数值模拟方法的各种三维波动问题。此外,构造了3.5维波动方程的一个求解基准,为四维空间波传播特性的研究打开了局面。
The wave equation with semi-integer dimension embodies an abstract way of thinking from point to line,from line to surface,and from surface to volume.It can reveal the propagation characteristics of various forms of waves in the space at a lower cost and a higher efficiency.However,its inherent mechanism is relatively complex,and it requires a high level of detail control,so it has considerably high technological threshold in obtaining its solution.The finite element method in the frequency domain is taken as an example.Started from the deduction of 1.5-Dimensional acoustic wave equation,elastic wave equation and their coupling form,the technological details such as the selection of stretching scale of Perfectly Matched Layer(PML),the derivation of weak form of equivalent integral,the setting of coupling boundary conditions and the avoidance of poles are noted.The 1.5-Dimensional wave equation coupled with acoustic wave and elastic wave is used to solve the 2-Dimensional issues.The numerical solutions of the finite element method in frequency domain are in good agreement with the analytical solutions.Such methods of thinking can be easily extended to various 3-Dimensional wave propagation issues characterized by the 2.5-Dimensional wave equation based on other numerical simulation methods.In addition,a benchmark for solving the 3.5-Dimensional wave equation is established,which opens up a prospect for the study of wave propagation characteristics in the 4-Dimensional space.
作者
张阔
ZHANG Kuo(China Center for Information Industry Development,Beijing 100048,China;Beijing CCID Publishing&Media Co.,Ltd.,Beijing 100048,China)
出处
《新一代信息技术》
2019年第21期13-21,共9页
New Generation of Information Technology
基金
国家科技重大专项课题(项目编号:2011ZX05020)。
关键词
半整数维波动方程
频域有限元方法
声波与弹性波耦合
Wave equation with semi-integer-dimension
Finite element method in frequency domain
Coupling of acoustic wave and elastic wave
Perfectly matched layer
Weak form of equivalent integral
Pole
Notes
