当前位置: X-MOL 学术Multiscale Modeling Simul. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Hadamard--Babich Ansatz for Point-Source Elastic Wave Equations in Variable Media at High Frequencies
Multiscale Modeling and Simulation ( IF 1.9 ) Pub Date : 2021-01-13 , DOI: 10.1137/20m1322224
Jianliang Qian , Jian Song , Wangtao Lu , Robert Burridge

Multiscale Modeling &Simulation, Volume 19, Issue 1, Page 46-86, January 2021.
Starting from Hadamard's method, we develop Babich's ansatz for the frequency-domain point-source elastic wave equations in an inhomogeneous medium in the high-frequency regime. First, we develop a novel asymptotic series, dubbed Hadamard's ansatz, to form the fundamental solution of the Cauchy problem for the time-domain point-source elastic wave equations in the region close to the source. Using the properties of generalized functions, we derive governing equations for the unknown asymptotics of the ansatz including the travel time functions and dyadic coefficients. In order to derive the initial data of the unknowns at the point source, we further propose a condition for matching Hadamard's ansatz with the homogeneous-medium fundamental solution at the point source. To treat singularity of dyadic coefficients at the source, we then introduce smoother dyadic coefficients. Directly taking the Fourier transform of Hadamard's ansatz in time, we obtain a new ansatz, dubbed Hadamard--Babich ansatz, for the frequency-domain point-source elastic wave equations. To verify the feasibility of the new ansatz, we truncate the ansatz to keep only the first two terms, and we further develop partial-differential-equation--based Eulerian approaches to compute the resulting asymptotic solutions. Numerical examples demonstrate the accuracy of our method.


中文翻译:

Hadamard--Babich Ansatz 用于高频可变介质中的点源弹性波方程

多尺度建模与仿真,第 19 卷,第 1 期,第 46-86 页,2021 年 1 月。
从 Hadamard 的方法开始,我们为高频区域的非均匀介质中的频域点源弹性波方程开发了 Babich 的 ansatz。首先,我们开发了一个新的渐近级数,称为 Hadamard 的 ansatz,以形成靠近源的区域中时域点源弹性波方程的柯西问题的基本解。利用广义函数的性质,我们推导出了 ansatz 的未知渐近线的控制方程,包括走时函数和二进系数。为了推导出点源处未知量的初始数据,我们进一步提出了将Hadamard's ansatz与点源处的均质介质基本解相匹配的条件。为了在源处处理二进系数的奇异性,然后我们引入更平滑的二进系数。直接及时对Hadamard ansatz进行傅里叶变换,我们得到了一个新的ansatz,称为Hadamard--Babich ansatz,用于频域点源弹性波方程。为了验证新 ansatz 的可行性,我们截断 ansatz 以仅保留前两项,并且我们进一步开发了基于偏微分方程的欧拉方法来计算所得渐近解。数值例子证明了我们方法的准确性。我们截断 ansatz 以仅保留前两项,并进一步开发基于偏微分方程的欧拉方法来计算所得渐近解。数值例子证明了我们方法的准确性。我们截断 ansatz 以仅保留前两项,并进一步开发基于偏微分方程的欧拉方法来计算所得渐近解。数值例子证明了我们方法的准确性。
更新日期:2021-01-13
down
wechat
bug