波动的有限元模拟——基本问题和基本研究方法

FINITE ELEMENT SIMULATION OF WAVE MOTION ——Basic Problems and Conceptual Aspects

本文以紧凑形式给出了可以直接用于波动有限元计算的透射人工边界公式。为了阐明离散化误差和由人工边界引起的振荡失稳的基本概念,作者利用简单的一维模型详细分析了波动有限元模拟中遇到的这两个主要问题。基于在离散和连续模型中波动规律差别的分析,首先研究了离散化误差。这一研究导致对可能实现波动有限元模拟的频段的识别,以及采用一种兼备有限差和有限元优点的集中质量离散模型的建议。根据波在人工边界的放大效应和在有限离散模型中多次反射的概念在频域内阐明了这一常见的振荡失稳的机制。在此基础上提出了在保证数值计算精度的条件下通过对人工边界附近节点运动的修正来消除振荡失稳的措施。本文还以一维模型为例给出了人工边界条件的稳定性准则。

The artificial boundary condition is written in a compact form, which can be djrectly incorporated into finite elements. The discretization error and the oscillation instability are studied in detail for a simple one-dimensional model in order to clarify conceptual aspects of the two basic problems in the simulation. The discretization error is analyzed through studies on characteristics of wave motion in the discrete model and their differences from those in the corresponding continuum. The analysis leads to identifying a frequency band within which the simulation is possible,and to a suggestion of using the lumped-mass finite element model for the simulation. Mechanism of the oscillation instability is then illuminated in frequency domain by amplification at the artificial boundary and multi-reflection of wave motion in a finite discrete model. Understanding of the mechanism provides guidelines for seeking measures to guarantee the stable implementation of the artificial boundary, and a modification of the artificial boundary condition is then devised for eliminating the instability. The special stability criterion is finally presented for the modified boundary condition, in addition to the ordinary stability condition for the numerical integration of equation of motion at interior nodes.