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多分辨率平面单元有限元法

发表时间:2016-01-15  浏览量:3308  下载量:873
全部作者: 夏逸鸣
作者单位: 南京航空航天大学航空宇航学院
摘 要: 基于传统平面应力单元节点形函数,通过对形函数关于坐标零点的延拓,形成单元的基本节点形函数;并将基本节点形函数在单元区域上进行伸缩和平移,由此生成相互嵌套、逐级包含的位移子空间序列,即多分辨分析(multiresolation analysis,MRA)的简单而明晰的数学基础。由此,建立了带有分辨率常数的理性多辨分析的概念;并在此基础上,构建多分辨率平面单元和相应的有限元法。该单元为无网格单元,具有如下特性:1)可通过自由调节单元分辨率(resolution level,RL)的大小来增减单元内的节点数量,从而调整单元计算的清晰度,结构整体分析可由数个多分辨率单元经拼接而实现,因此,结构分析的清晰度是由分辨率决定的而不是网格;2)经重新排布节点后的单元整体刚度、等效节点荷载向量可按节点直接获得,不像传统单元那样需要人工重新生成;3)与传统单元一样可以很方便地处理各种边界条件。因此,传统平面单元有限元方法本质上是一种单分辨率单元有限元方法,为本文多分辨率单元有限元方法的特例。
关 键 词: 应用数学;平面单元;基本节点形函数;伸缩和平移;位移子空间序列;理性多辨分析
Title: A multiresolution finite element method based on a new plane element
Author: XIA Yiming
Organization: College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics
Abstract: Based on a node shape function for the plane stress element, the basic node shape function is constructed by extending the node shape function to other three quadrants around the coordinate zero point and a mutually nesting displacement subspace sequence is constituted out of scaling and shifting of the basic node shape function on the element domain, which brings about a simple, clear and rigorous mathematical basis for the multiresolution analysis (MRA). After that, the multiresolution plane stress element and method is formulated through applying the minimum potential energy principle. Hence, the MRA concept is established and the resolution level (RL) for the element is introduced. Hence, the proposed element is a meshless one and possesses the following features: 1) the node number the element contains can be modulated by adjusting the RL, thus improving the element analysis clarity. The overall analysis of a structure can be executed through splicing elements, hence the analysis clarity being actually determined by the RL, not by the mesh. 2) The element stiffness matrix, node equivalent loading vector can be automatically acquired for the proposed while those being obtained by artificially reassembling for the traditional after the node layout is reset. 3) The boundary conditions can be conveniently handled by the proposed just as those by the traditional. As a result, traditional 4-node plane stress element and method is essentially a mono-resolution one and also a special case of the proposed element and method.
Key words: applied mathematics; plane element; basic node shape function; scaling and shifting; mutually nesting displacement subspace sequence; rational multiresolution analysis
发表期数: 2016年1月第1期
引用格式: 夏逸鸣. 多分辨率平面单元有限元法[J]. 中国科技论文在线精品论文,2016,9(1):93-100.
 
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