Study On High-order Doubly Asymptotic Transmitting Boundary And Its Application

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.