完全匹配层在矩阵式波动方程SBP-SAT方法应用

Application of perfect matching layer in SBP-SAT method of matrix wave equation

数值离散方法和截断边界效果是地震动模拟实现的关键。基于分部求和(summation-by-parts,SBP)和一致逼近(simultaneous approximation term,SAT)的SBP-SAT方法具有较高的稳定性,这使得该方法具备了较高的应用前景和价值。此外,完全匹配层(perfect matching layer,PML)是一种应用广泛用于模拟截断边界的技术,但引入匹配层可能会破坏原始方程的稳定性,特别是在各向异性介质或曲线域模型中。首先基于数理推导,给出弹性波动方程系数矩阵的对称形式。在此基础上,引入多轴完全匹配层(multi-axis perfect matching layer,MPML),并建立相应的匹配层方程。通过本征值分析,我们可以判断阻尼函数对原方程特征根实部的走向和取值范围的影响。然后,我们采用SBP-SAT方法对矩阵对称形式匹配层方程进行离散,并在频域中采用能量法进行稳定性评估。通过对不同模型的数值仿真,表明所提出的离散框架具有整合度高、稳定性好和拓展性强等特点。此外,多轴匹配层可以与SBP-SAT方法结合,可以稳定地模拟曲线域中的波传播。

Numerical discretization method and truncation boundary effect are key to realization of ground motion simulation.SBP-SAT method based on summation-by-parts(SBP)and simultaneous approximation term(SAT)has higher stability to make it have higher application prospects and value.In addition,the perfect matching layer(PML)is a widely used technique for simulating truncated boundaries,but introducing matching layer may destroy the stability of the original equation,especially,in anisotropic media or curved domain models.Here,firstly,based on mathematical and physical derivation,the symmetric form of coefficient matrix of elastic wave equation was deduced.Then,the multi-axis perfect matching layer(MPML)was introduced and the corresponding matching layer equation was established.Through eigenvalue analysis,effects of damping function on trend and value range for real part of eigenvalues of the original equation were judged.Furthermore,SBP-SAT method was used to discretize the matching layer equation in the form of symmetric matrix,and the energy method was used to evaluate its stability in frequency domain.Numerical simulation results of different models showed that the proposed discrete framework has characteristics of high integration,good stability and strong scalability;the multi-axis matching layer can be combined with SBP-SAT method to stably simulate wave propagation in curved domain.