完全匹配层中衰减函数的参数优化分析

完全匹配层(perfectly matched layer,PML)是一种高效的吸收边界条件,对体波和面波都有非常好的吸收效果,被广泛应用于弹性波数值模拟。针对二维弹性动力学问题,基于PML最大反射系数的理论推导,对PML衰减函数中的参数取值进行了优化分析。首先基于复伸展坐标变换,给出了一种适用于二阶弹性波动方程的非分裂吸收边界条件;然后通过求解平面P-SV波的波数,得到了非分裂完全匹配层反射系数的解析表达式;最后采用COLLINO给出的衰减函数形式,令PML最大反射系数为最小,得到了衰减函数中PML的厚度、理论反射系数以及最大反射系数之间的相互关系。通过数值算例分析了PML最大反射系数的变化规律,为PML参数的选择和优化提供了理论依据。

Perfectly matched layer (PML) is a high-efficiency absorbing boundary condition (ABC) to body waves and surface waves,and is widely applied to the numerical simulation of the elastic wave.Considering the two-dimensional elastic dynamic problems,on the basis of the theoretical solution of PML’s maximum reflection coefficient,we carry out optimization analysis on the parameters selection for the attenuation function in the PML.Primarily,based on the complex-stretching-coordinate transform,we deduced one kind of the unsplit absorbing boundary condition appropriate for the second-order elastic wave equation.Then by solving the wave number of plane P-SV waves,we obtained the analytical expression of maximum reflection coefficient of the PML.We chose the attenuation function form proposed by Francis Collino,by ordering the maximum reflection coefficient approaching the minimum value,and eventually obtained the mutual relations of the PML’ s thickness,the theoretical reflection coefficient and the maximum reflection coefficient.Finally,we analyzed the variation rule of PML’ s maximum reflection coefficient by numerical testing,which provided theoretical guidance for the selection and optimization of PML’ s parameters.