黏弹性多层介质中SH波动的一种吸收边界条件  

AN ABSORBING BOUNDARY CONDITION FOR SH WAVE PROPAGATION IN VISCOELASTIC MULTILAYERED MEDIA

作　　者：吴利华[1] 赵密[1] 杜修力[1] Wu Lihua;Zhao Mi;Du Xiuli(Key Laboratory of Urban Securiy and Disaser Engineering of Minisry of Education,Beijing University of Technology,Beiing 100124,China)

机构地区：[1]北京工业大学城市与工程安全减灾教育部重点实验室,北京100124

出　　处：《力学学报》2020年第2期480-490,共11页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家重点研发项目(2018YFC1504305);国家自然科学基金项目(51678015);中国教育部创新团队(IRT 17R03)资助。

摘　　要：提出一种高精度时域吸收边界条件,与有限元法结合用于模拟瞬态标量SH波在达朗贝尔黏弹性多层介质中传播问题.建立时域吸收边界条件的过程是:首先将半无限域沿着竖向半离散得到半离散的位移方程以及人工边界处的力-位移关系,再通过引入模态分解,将物理空间下的量转化到模态空间,从而获得半无限域模态空间下的频域动力刚度;其次采用一种在全频范围内收敛的连分式精确逼近单层介质模态空间下标量形式的频域动力刚度,将标量连分式扩展为矩阵形式用来表示多层介质的频域动力刚度;最后通过引入辅助变量技术,将模态空间下基于连分式的频域动力刚度关系转化为时域吸收边界条件,进一步转换到物理空间后得到物理空间下的时域吸收边界条件.单层介质和五层介质的数值算例表明,建立的高精度时域吸收边界条件对于达朗贝尔黏弹性单层介质是精确且稳定的;对于达朗贝尔黏弹性多层介质,为了保证其高精度特性,需要将人工边界放置在距离感兴趣区域约为0.5倍无限域高度的位置处.A high-accuracy absorbing boundary condition in the time domain is proposed, which can be coupled with the finite element method seamlessly to simulate the propagation of the transient scalar wave in the D’Alembert viscoelastic multilayered media. First, a semi-discrete displacement equation of the semi-infinite domain and the force-displacement relationship in the artificial boundary are obtained by semi-discretizing the semi-infinite domain along the vertical depth.Modal decomposition is utilized to convert the field of the semi-infinite domain in the physical space into the modal space.Then the dynamic stiffness of the semi-infinite domain in the frequency domain in the modal space is obtained according to both the displacement equation and the force-displacement relationship in the modal space. Second, a scalar continued fraction, which is convergent over the whole frequency domain, is proposed to describe the scalar dynamic stiffness in the modal space of the D’Alembert viscoelastic single-layered medium. The scalar continued fraction is extended to the matrix form to represent the dynamic stiffness in the modal space of the D’Alembert viscoelastic multilayered media.Last, by introducing auxiliary variables, a time-domain absorbing boundary condition in the modal space is constructed based on the proposed continued fraction. Subsequently, considering the relationship of the field in the modal space and in the physical space, a time-domain absorbing boundary condition in the physical space is obtained by converting the absorbing boundary condition in the modal space into the physical space. Two numerical examples of a single-layered medium and a five-layered media verify that the proposed method is accurate and stable for the D’Alembert viscoelastic single-layered medium, and for the D’Alembert viscoelastic multilayered media, in order to ensure the proposed method’s property of high-accuracy, the distance from artificial boundary to the region of interest needs to be about 0.5 times of the total

关 键 词：高精度吸收边界条件 连分式 达朗贝尔黏弹性多层介质 标量波 有限元 

分 类 号：O241[理学—计算数学]