透射边界高频耦合失稳机理及稳定实现--SH波动

Mechanism of high frequency coupling instability and stable implementation for transmitting boundary——SH wave motion

就大型近场波动的高效数值模拟而言,稳定实现高阶人工边界是一个尚未圆满解决的问题.本文针对使用多次透射公式的SH波动集中质量有限元模拟,依据GKS定理的群速度解释,进一步阐明了人工边界与内域离散格式耦合所导致高频失稳的机理,即两者支持群速度指向内域的外行高频平面谐波,波动能量自发地从人工边界进入內域,从而导致失稳,而这类谐波是由集中质量有限元离散引入的.本文提出了消除此种耦合失稳的一种方法:通过修改有限元刚度阵来改变内域离散格式,并保证修改格式的精度不低于原有格式的精度.理论分析和数值实验表明此法能稳定实现透射边界.本文研究结果具有推广应用前景.

Stable implementation of high order artificial boundary conditions has not been solved satisfactorily for wave simulation in infinite domain.The instability problem is studied for the multi-transmitting formula,one of the promising boundary schemes.Focusing on the lumpedmass finite element simulation of SH wave motion,mechanism of high frequency instability caused by coupling the formula and the finite element scheme is further clarified based on the group velocity interpretation of GKS theorem.The mechanism is that both the formula and the finite element scheme support out-going high frequency waves whose group velocity points to interior domain,thus wave energy is allowed to enter spontaneously from the boundary into interior domain.Such type of waves do not exist in the corresponding continuous model,it is solely induced by the finite element.A method for destroying the coupling is presented by changing the interior scheme via modifying the stiffness matrix of FEM using proper integrationrule,meanwhile the accuracy of the modified scheme is no less than that of the original one.Theoretical analysis and numerical experiments have shown this method can ensure MTF to be implemented stably.The research approach developed in this study can be applied to cope with the boundary instable problems for other types of wave simulation,and the research results have application prospect for forward and inverse problems in geophysics.