采用黏弹性人工边界单元时显式算法稳定性的改善研究

IMPROVEMENT OF EXPLICIT ALGORITHMS STABILITY WITH VISCO-ELASTIC ARTIFICIAL BOUNDARY ELEMENTS

黏弹性人工边界单元是目前常用的处理半无限空间波动问题的数值模拟方法,可有效吸收计算区域内产生的外行波动.黏弹性人工边界单元具有与内部介质不同的质量密度、刚度和阻尼,受其影响,对整体模型进行显式时域逐步积分时,在边界区域易发生失稳现象,影响整体系统显式积分的计算效率.针对该问题目前尚无行之有效的解决方法.本文针对二维黏弹性人工边界单元,建立可代表整体系统典型特征的侧边子系统和角点子系统,利用传递矩阵谱半径分析方法,基于传统中心差分格式,推导得到局部子系统稳定性条件的解析解.在此基础上通过研究解析解中各物理参数对稳定性条件的影响,给出通过增加人工边界单元的质量密度,以改善采用黏弹性人工边界单元时显式算法稳定性的方法.均匀和成层半空间波动问题算例分析表明,将内部单元质量密度设置为人工边界单元质量密度的上限,可以在保证黏弹性人工边界计算精度的前提下,有效改善整体系统显式时域逐步积分的数值稳定性,大幅提高计算效率.

Viscoelastic artificial boundary elements are commonly applied in the analysis of semi-infinite wave propagation problems,which can accurately absorb the scattered waves generated in the calculation domain.However,the mass density,the stiffness and the damping coefficient of the viscoelastic artificial boundary element are different from those of the internal domain.Therefore,the instability often occurs in the boundary region when the explicit time-domain stepwise integration is performed in the overall model,so,the calculation efficiency of the explicit integral for the overall system is affected.Currently,there is no effective solution to this problem which remains to be settled to conduct efficient large-scale wave propagation simulation.For the two dimensional viscoelastic artificial boundary element,we establish the edge subsystem and corner subsystem which can represent the typical characteristics of the overall system,by the analysis method of transfer matrix spectral radius based on the central difference format commonly used,we derive the analytical solutions of the stability conditions of the edge subsystem and corner subsystem.After that,we analyze the influence of various physical parameters of the two dimensional viscoelastic artificial boundary element on the stability conditions,and obtain the method for improving the stability condition of the explicit algorithm by increasing the mass density of the viscoelastic artificial boundary element.The homogeneous and layered half-space examples show that,set the mass density of the internal element as the upper limit of the mass density of the viscoelastic artificial boundary element,the method proposed in this paper can effectively improve the numerical stability of the explicit time-domain integration when using the viscoelastic artificial boundary elements,without affecting the calculation accuracy,and the calculation efficiency can be significantly improved in the explicit dynamic analysis.The model size(distance from the scattered wave source to the artificial boundary)has a little effect on the stability of the explicit integral which can be ignored.