层状介质波动问题的高阶双渐近透射边界研究及应用

Accurate modeling the radiation damping of unbounded reservoir and foundation is a key topic in dynamic dam-reservoir-foundation interaction analysis in earthquake engineering. The propagation of vector waves in a semi-infinite layer is addressed by using the scaled boundary finite element method. The main contents are summarized as follows: (1) a high-order doubly asymptotic open boundary is constructed based on the continued fraction solution of the dynamic stiffness matrix. This open boundary is expressed as a system of firs-order ordinary differential equations in the time domain. To evaluate the numerical stability of the proposed open boundary, the generalized eigenvalue problem for the coefficient matrices is analyzed. Possible spurious modes can be eliminated using the spectral shifting technique. (2) To avoid the numerical instability, the vector waver equation is decoupled into individual scalar waver equations. The high-order doubly asymptotic open boundary for scalar wave is used to approximate the vector wave propagation. (3) Embedding the high-order asymptotic open boundary directly into the finite element scheme, a seamless time-domain coupled model for dynamic dam-reservoir-foundation interaction analysis is established. The dam-reservoir-foundation system is divided into two parts: the near field is modeled by the finite element method, the far field, namely the semi-infinite reservoir and foundation is modeled by the excellent high-order doubly asymptotic open boundary. Employing the suitable seismic input mechanism, the seismic responses of practical engineering dam-reservoir-foundation systems subjected to strong earthquakes are investigated. The effects of radiation damping of unbounded medium are further studied. The main findings of this project will make important contribution to the development of the method for seismic response analysis of hydraulic structures.

无限库水和无限地基辐射阻尼的模拟是大坝-库水-地基动力相互作用分析的核心问题之一。本项目针对半无限层状介质矢量波传播问题，在比例边界有限单元法的理论框架下主要开展以下几个方面的研究工作：（1）基于动力刚度连分式渐近解构建矢量波传播模拟的高阶双渐近透射边界，采用广义特征值分解的方法分析其数值稳定性，并通过移谱法消除导致数值不稳定性的虚假模态。（2）将矢量波方程强行解耦成独立的标量波方程，发展一种简化的模拟矢量波传播的高阶双渐近透射边界。（3）将该透射边界同有限元法结合，提出一种新的坝-库水-地基动力相互作用分析的时域计算模型，其中近场有限域采用有限元法模拟，远场无限库水和无限地基采用高阶双渐近透射边界模拟；结合相适配的地震输入机制，研究大坝在强震作用下的动力响应，进一步揭示无限介质辐射阻尼的影响。本项目的研究成果将对水工结构抗震分析方法的发展提供必要的理论依据。

大坝与半无限库水、半无限地基之间的动力相互作用是大坝地震响应分析应当考虑的重要方面，而波动在无限介质中传播引起的辐射阻尼效应模拟是该问题的重点和难点。高阶双渐近透射边界具有在全频范围内收敛到动力刚度准确解的优点，具有很高的计算精度和计算效率。本项目针对层状介质波传播问题，在比例边界有限单元法的理论框架下开展高阶双渐近透射边界研究，主要研究工作包括以下几个方面：（1）针对二维层状介质弹性波问题，结合动力刚度改进连分式双渐近解和辅助变量技术构建了高阶双渐近透射边界；基于广义特征值分解分析其数值稳定性，并通过移谱法消除导致数值不稳定的虚假模态；数值算例表明该透射边界具有很高的计算精度和计算效率。（2）将二维弹性动力学方程强行解耦得到两个独立的标量波动方程，采用标量波高阶双渐近透射边界近似模拟矢量波在层状中的传播；数值算例表明该简化高阶双渐近透射边界对底部为固定边界条件的层状介质具有较高的计算精度，并且具有良好是数值稳定性。（3）结合有限元法和高阶双渐近透射边界，提出了一种大坝-库水-层状地基动力相互作用分析的时域计算模型，其中大坝和近场库水、地基采用有限单元离散，远场半无限库水和地基采用高阶双渐近透射边界模拟；将自由场地震动转化为截断边界等效节点力以实现地震输入，并进行了大坝动力响应分析；算例分析结果表明该时域耦合分析模型具有很高的计算精度和较高的计算效率，适用于实际大坝长时间的地震响应分析。本项目提出的高阶双渐近透射边界是一种精度和效率兼顾的数值方法，对无限层状介质波传播模拟研究具有重要的理论意义和一定的工程实用价值，可为我国大坝地震响应分析方法的发展提供必要的理论依据。