无限元方法及其应用

限元是几何上趋于无穷的单元，它是一种特殊的有限元,也是对有限元在求解无界域 问题上的有效补充, 并可实现与有限元间的无缝连接.无限元分为映射无限元和非映射 无限元:映射无限元需要引入几何映射,在局部坐标系中构造插值形状函数,如Bettess 元和Astley元;非映射无限元则直接在整体坐标系中构造插值形状函数,如Burnett元. 本文评述求解无界域问题的无限元方法的研究现状和最新发展.首先介绍无限单元的概念 和无限元方法的特点;围绕求解以Helmholtz方程控制的波动问题,评述几种常规无限单 元的优劣,这些单元包括Bettess元、Astley元和Burnett元.然后介绍新近提出的广义 无限元方法,以及与常规无限元方法的区别与联系.最后对无限元方法在各种问题中的 应用做了总结.

THE INFINITE ELEMENT METHOD AND ITS APPLICATION

An infinite element is the one that can handle a domain of infinity.It is a special finite element, and can have a seamless connection with convertional finite elements.It can be a mapped infinite element or a non-mapped infinite element.The former,such as Bettess element and Astley element,needs geometry mapping and shape functions in terms of local coordinates,while for the latter,the shape functions are directly expressed in terms of global coordinates.This paper reviews the state-of-the-art and recent advances of the infinite element method for unbounded domains.First,the concept and features of the infinite element method are introduced.Then,taking the wave problems governed by the Helmholtz equation as an example,several conventional infinite elements such as the Bettess element,the Astley element and the Burnett element are compared and reviewed.Next,we introduce the generalized infinite element and its relation to the conventional infinite elements.Finally,the applications of the infinite element to various problems are summarized.