一维波动有限元模拟中透射边界的时域稳定条件

Time-domain Stability Conditions of Multi-transmitting Formula in One-dimensional Finite-element Simulation of Wave Motion

多次透射边界是近场波动数值模拟的一种重要人工边界条件,但其数值稳定性问题一直是致力于解决的关键问题,同时亦是研究难点.本文提出了透射边界时域稳定条件的概念,从透射边界时域计算参数方面界定边界稳定性特征.基于空间插值模式讨论了波动有限元模拟中透射边界可能采用的数值计算格式,进而根据频域反射系数和反射放大失稳准则导出明确的透射边界时域稳定条件.定量地研究了一阶和高阶透射边界不同数值计算格式下人工波速取值和透射阶次对边界稳定性的影响,结果显示人工波速取值远超过介质物理波速或高阶边界选择了不恰当的数值计算格式是导致边界失稳的重要原因.将透射边界和经典的Clayton-Engquist边界、Higdon边界等统一归结为位移型人工边界范畴,其中透射边界具有可调人工波速和不同透射阶次的参数配置,概括了这类边界的基本特征.提出借鉴经典位移型人工边界的研究成果如箱型计算格式、纵横波速比限值、调整边界表达形式等,来改善透射边界时域稳定性的建议.

Multi-transmitting formula(MTF)is an important artificial boundary condition for numerical simulation of near-field wave motion,but its numerical stability is a key problem that hasn’t been completely resolved.The concept of time-domain stability conditions of multi-transmitting formula is proposed in this work,which indicates that computation parameters are highly relevant to the stability of the boundary.Alternative numerical schemes of multi-transmitting formula based on spatial interpolation are systematically discussed for the application in finite-element simulation.Frequency-domain reflection coefficients and the criterion of reflection-amplification instability are used to obtain rigorous time-domain stability conditions of multi-transmitting formula,which reveals that instability may ocuurs when artificial wave speed far more than physical wave velocity or inappropriate numerical schemes of high-order boundaries are used.Multi-transmitting formula and classical Clayton-Engquist boundaries,Higdon boundaries et al are uniformly considered as displacement-type artificial boundary condition,which has two basic parameters similar to that of the MTF.Thus,the stability of these boundary conditisons is comparable,and it is recommended that research findings of other classical displacement-type boundary conditions,such as box-type computational schemes,thresholds of the aspect wave velocity ratio,different boundary formulations et al,can be taken into account to improve the time-domain stability of multi-transmitting formula in numerical simulation of wave motion.