弱形式时域完美匹配层——滞弹性近场波动数值模拟

Weak-form time-domain perfectly matched layer for numerical simulation of viscoelastic wave propagation in infinite-domain

本文旨在构建适用于滞弹性近场时域波动有限元模拟的高精度人工边界条件:完美匹配层(Perfectly Matched Layer:PML),其中阻尼介质时域本构基于广义标准线性体建立.与以往研究不同,本文采用复坐标延拓技术变换弱形式波动方程构建了可直接用有限元离散的弱形式时域PML,规避以往独立对无限域内波动方程及界面条件进行延拓可导致的PML场方程和界面条件匹配不合理引发数值失稳、计算精度低下等问题.其次,针对PML中多极点有理分式与频域函数乘积的傅里叶反变换难以计算的问题,利用PML精度对复坐标延拓函数中延拓参数微调不敏感这一特点,明确给出了参数微调准则以规避多重极点,进而利用有理分式分解给出了一种普适、简便的计算方法,极大地简化了PML计算.基于该方法可实现任意高阶PML.最后,将本文构建滞弹性PML与高阶勒让德谱元(高精度集中质量有限元)结合得到滞弹性近场波动谱元离散方案.基于算例验证了滞弹性PML的计算效率、精度及新离散方案的长持时稳定特性.新离散方案可应用于计入实际介质阻尼的地震波动正、反问题数值模拟,提高波形模拟的精度以及地下波速结构反演的精度和可靠性.

Taking into account of the viscoelastic effect of a damping medium in the infinite domain,a novel viscoelastic PML(perfectly matched layer)for truncation of infinite-domain has been constructed,inside which the constitutive relation of the damping medium is established based upon the generalized standard linear body.Being different from previous studies,the PML in a weak-form is derived by complex stretching the weak-form viscoelastic wave equation in the infinite domain.Compared with the strong-form PML derived in a traditional way,inside which the field equation of PML together with the boundary and/or interface condition of PML are independently obtained by complex stretching their counterparts in infinite domain,the mismatch between the obtained field equation and boundary and/or interface condition can be naturally avoided.Secondly,a clean and simple approach for time-domain PML implementation has been proposed to avoid the difficult computation for inverse Fourier transform of the complex multi-pole rational fraction kernels.To accomplish this,we utilize the feature that the accuracy of PML is insensitive to slightly tuning of the parameters in the complex stretching function to decompose such kernels into the sum of single-pole fraction kernels,of which the corresponding convolution integral can be localized and computed in an iterative way.The new approach facilitates the implementation of arbitrary high-order PML.Finally,utilizing the novel viscoelastic PML for infinite domain truncation,a new scheme for infinite-domain viscoelastic wave simulation has been derived using high-order spectral element methods for space discretization.The computational efficiency and accuracy of the novel viscoelastic PML and the long-term stability of the new scheme have been verified by numerical tests.The new scheme can also be used in full waveform inversion taking into account of the damping effect of the real medium to improve the accuracy and robustness of inversion for the underground wave velocity structure.