弱形式时域完美匹配层

摘要

In near-field wave simulation,it is necessary to truncate the infinite-domain via highly accurate absorbing boundary condition to improve the computational efficiency.Perfectly matched layer is one kind of highly accurate absorbing boundary condition formulated as absorbing layer.In traditional way,the strong-form field equation of PML together with the boundary and/or interface condition of PML are obtained by complex stretching their counterparts in infinite domain.However,the boundary and/or interface condition of infinite domain have also been applied for PML without any modification.The construction of PML's field equation and its boundary and/or interface conditions are independent of each other,the two may be improperly matched,which lead to numerical instability and the deterioration of numerical accuracy.In this paper,a new method for PML derivation is proposed.PML is obtained by complex stretching the weak form wave equation in infinite domain.Since the weak-form wave equation has combined the wave equation and the boundary and/or interface condition,the mismatch between the obtained field equation and boundary and/or interface condition can be naturally avoided.Via the new method,the weak-form time-domain PML can be derived in a straight way,strong-form PML can also be derived.The former is ready for discretization with finite element method,while the latter could be discretized by finite difference method.Applying Legendre spectral element method for space discretization,full scheme for near-field wave simulation in elastic media has been established.The numerical stability and the accuracy of new scheme is illustrated by numerical tests.The method for PML construction can be directly applied for multi-phase media near-field wave simulation.%应用高精度人工边界条件可有效提升近场波动数值模拟计算效率.完美匹配层是吸收层形式高精度人工边界条件,匹配层内场方程和界面条件通常分别采用复坐标延伸技术变换强形式无限域内波动方程和界面条件得到,亦曾将无限域界面条件当作匹配层界面条件.场方程和界面条件构建过程相互独立,可能出现匹配不合理而引发数值失稳、计算精度低下等问题.本文提出采用复坐标延伸技术变换弱形式无限域波动方程以构建完美匹配层的方法.弱形式波动方程耦合了波动方程及界面条件,进而规避了变换后所得场方程与界面条件之间的匹配不合理问题.新方法可直接建立弱形式匹配层,在此基础上亦可给出强形式匹配层.弱形式便于有限元离散,强形式便于有限差分离散.基于弱形式完美匹配层,结合勒让德谱元建立了弹性介质近场波动谱元模拟方案.利用算例验证了新方案的精度及数值稳定性.本文工作可直接推广至多相耦合介质近场波动数值模拟.