【作者】 高毅超；

【导师】 金峰；

【作者基本信息】 清华大学 ， 水利工程， 2013， 博士

【摘要】 基于比例边界有限单元法发展的高阶透射边界是具有良好计算性能的无限域数值模拟方法。本文从比例边界有限单元法的控制方程出发，致力于发展可用于标量波问题和矢量波问题模拟的高阶双渐近透射边界，并同有限单元法结合，在应用方面进行了探索和研究。论文的工作主要包括以下几个方面：1.基于开源软件框架OpenSees数值实现了动水压力波高阶双渐近透射边界与有限单元法的直接耦合分析模型和分离耦合分析模型，并用于大坝-库水动力相互作用分析。数值分析结果表明两种耦合分析模型均具有很高的计算精度和计算效率，在实际应用中可以选取较少的透射边界阶数、减少近场有限单元离散的范围，进一步提高求解效率。2.通过引入简单的反射系数考虑库底柔性，基于比例边界坐标变换和伽辽金加权余量法推导了三维层状库水的比例边界有限单元方程，并构建了动力刚度的改进连分式双渐近解。通过引入辅助变量发展了可以考虑库底柔性的高阶双渐近透射边界，并将其嵌入到近场有限单元方程中，用于大坝-库水动力相互作用分析。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。3.针对二维半无限水平层状介质，将弹性动力学方程强行解耦得到两个与标量波方程形式相同的波动方程，发展了一种简化的模拟层状介质矢量波的高阶双渐近透射边界。数值算例表明，该高阶透射边界对于底部为固定边界条件的水平层状介质具有良好的计算精度；它具有与标量波高阶双渐近透射边界相同的计算效率以及良好的数值稳定性。4.从弹性动力学的比例边界有限单元方程出发推导了动力刚度的改进连分式双渐近解，发展了高阶双渐近透射边界。为了获得准确的计算结果，论文利用基函数的性质，将比例边界有限单元方程拆分成高阶模态方程和低阶模态方程两部分，然后分别采用连分式双渐近解和高频渐近解求解。相比于高频连分式单向渐近解，该方法在低频段能够更快地收敛到准确解，因此在应用中可以选取较少的渐近阶数，从而提高计算效率。更多还原

【Abstract】 The high-order transmitting boundary based on the scaled boundary finite elementmethod is of excellent computational performance. The objective of this paper is todevelop high-order doubly asymptotic transmitting boundaries for modeling theunbounded domains. These high-order transmitting boundaries are applicable to bothscalar and vector waves and can be coupled seamlessly with finite element method. Themain contents are summarized as follows:1. Two coupled numerical methods for dam-reservoir interaction analysis aredeveloped by incorporating the excellent high-order doubly asymptotic transmittingboundary and the finite element method, namely the direct coupled method and thepartitioned coupled method. These two coupled methods are numerically implementedin the open-source finite element code OpenSees. Numerical experiments demonstratedthe high efficiency and accuracy of both coupled methods. As for the high accuracy ofthe high-order transmitting boundary, its order and the region discretized by finiteelement can be reduced in the analysis.2. The reservoir absorption effect is taken into consideration by introducing thereflection coefficient. Based on the scale boundary transformation of geometry and theGalerkin’s weighted residual technique, the scaled boundary finite element (SBFE)equation for three-dimensional layered reservoir is derived. Then, the improved doublyasymptotic continued fraction solution of the dynamic stiffness is constructed. Thecoefficient matrices are determined recursively from the SBFE equation in dynamicstiffness. By introducing the auxiliary variables, a high-order doubly asymptotictransmitting boundary is developed for modeling the unbounded reservoir. Thishigh-order transmitting boundary is incorporated with finite element method to analyzethe dam-reservoir interaction. Numerical examples demonstrate the high accuracy andefficiency of the proposed method.3. To simply the analysis of the two-dimensional semi-infinite layer with constantdepth, the governing elastic wave equation is decoupled into two scalar wave equationsby neglecting the coupling terms. As a result, the high-order doubly asymptotictransmitting open boundary developed for modeling the scalar wave can be applied tomodel the propagation of elastic waves in the semi-infinite layer. Numerical examples show that this method is applicable to the semi-infinite layer with fixed boundarycondition at the bottom. Besides, this method is of high computational efficiency andnumerical stability.4. A high-order doubly asymptotic transmitting boundary is developed to modelthe propagation of elastic waves in unbounded domains. This transmitting boundary isconstructed based on the improved doubly asymptotic continued fraction solution of thedynamic stiffness. To achieve accurate results, a set of weighted block-orthogonal basefunctions is introduced. The SBFE equation is split into a high-order mode equation anda low-order mode equation. The former is approximated by the doubly asymptoticcontinued fraction solution and the latter by the high-frequency singly asymptoticcontinued fraction solution. Compared to the high-order transmitting boundary based onthe high-frequency singly asymptotic continued fraction solution, the proposed methodhas a faster convergence speed in the low-frequency range. Thus, the order oftransmitting boundary can be reduced in the analysis to achieve higher computationalefficiency.更多还原