Stable Implementation of Transmitting Boundary in Numerical Simulation of Wave Motion

Abstract

The numerical instability resulted from the transmitting boundary in numerical simulation of the near‐field wave motion is systematically discussed from a unified viewpoint that wave energy within the computational region might accumulate and increase unceasingly. Mechanisms of two types of the instability, namely, the high‐frequency oscillation and the zero‐frequency drift are then clarified as follows: the former results from amplification of high‐frequency waves, which are meaningless for numerical simulation of wave motion by finite elements or finite differences, reflected from the transmitting boundary, and from repeated amplifications of multiple reflection of the waves within the finite computational region; the latter results from that the boundary allows the wave components of zero or approximately zero frequencies, entering the computational region continuously. Therefrom, a practical and complete scheme is suggested for stable implementation of the transmitting boundary that includes two simple measures: (1) injecting small amount of damping into the entire computational region by the procedure presented in the paper to eliminate the high‐frequency oscillation, and (2) using an operator algorithm of clear physical meaning to eliminate the zero‐frequency drift. The implementation scheme is finally tested in detail by numerical experiments of three‐dimensional wave motions including a source problem and a scattering problem.

Numerical simulation of water-bearing coal goaf using ground-borehole TEM—a case study of Tongxin Coal Mine, Shanxi Provice, China

“…where min (∆) is the minimum grid step size in the calculation model. On the ground-air boundary, the boundary area is approximately regarded as infinity, and the upward continuation method is used for processing; while selecting side and bottom boundary conditions, the modified Liao's absorption boundary conditions are used for calculation (Liao et al, 2002). This boundary condition handles the magnetic field at the boundary well and does not produce magnetic field distortion in the calculation area.…”

Analysis of the Three‐Dimensional Dynamic Problems by Using a New Numerical Method

“…From equations ( 3), (8), and (10), it can be seen that the tangential force on the surface (η, ζ), (ζ, ξ), and (ξ, η) of any given value ξ, η, and ζ is as follows:…”

“…Free or fixed boundary conditions are used to simulate infinite foundation soil. At the same time, in order to eliminate false reflection on artificial boundary, a transmission boundary model is established, for example, viscous and viscoelastic boundary (Zhang [6]), superposition boundary (Smith [7]), paraxial boundary (Engquist and Majda [8]), transient transmission boundary (Liao et al [9] and Liao et al [10]), and multidirectional and bi-asymptotic multidirectional transmission boundary (Keys [11] and Wolf and Song [12]). e main problems of finite element method based on the infinite boundary model, when it is used to analyze the dynamic consolidation of infinite foundation soil, are the inadequate accuracy of low-order boundary and the poor stability of high-order boundary.…”

A Fast Frequency Domain Method for Steady-State Solution of Forced Vibration of System with Complex Damping

“…For the forced vibration of system with complex damping, two methods are commonly used to obtain the steady-state responses of such a system, including the conventional frequency domain method (CFDM) and the dual-force-based time domain method (DTDM). The seismic response of soil layer and soil-structure interaction [10][11][12] are often calculated by the CFDM due to the complex damping of soil deposit. For example, the CFDM has been used to compute the nonlinear seismic response of one-dimensional soil layers [13,14], two-dimensional irregular-shaped basins with horizontal soil layers [15], and a hill in a layered elastic half space [16].…”