臺灣博碩士論文加值系統

有限元素特徵值問題在工程分析的領域中，具有極為重要的地位，而如何
善用分析方法成為一項重要的課題。本文的主旨即在研究各種有限元素特
徵值問題的解法，針對各種解法的執行速率和所得答案的精確度進行行探
討。在本研究中所針對的有限元素特徵值問題的解法，主要是自由度濃縮
法和特徵值直接解法兩個部份。而其中次結構是自由度濃縮法中最重要的
應用，次結構自由度濃縮法的主要用途，是將一個複雜而大型的結構體，
區分為若干個較小的次結構，如此可做到降低運算時間和記憶體需求量的
目的，另外更可方便於分析和設計工作。而在特徵值直接解法方面，分別
利用HQL(Householder-QL-inverse)法 、廣義賈可比法、子空間法和連佐
士法等四種方法，並針對各種解法的執行速率和所得答案的精確進行討論
，並由所得的結果，針對分析問題的維度大小，提出分析方法建議。本研
究並引用子空間法和連佐士法的混合模式─LS解法，探討此種混合模式在
提升執行速率和答案精確度的效果，而由其與子空間法和連佐士法的比較
得知，LS解法在執行速率上較子空間法為佳，且有良好的答案精確度和收
斂速度。所以LS解法是一項相當不錯的改良模式 。本文末，並以一工具
機之模態分析為實例，探討次結構自由度濃縮法應用於大型特徵值問題的
效果，如此可達到實例應用的目的。

The solution of the finite element eigenvalue problem has
become more complicated than before as the size and complexity
of the structure is increased. The objective of this paper is
to describe and to evaluate the performance of the methods for
the solution of the finite element eigenvalue problem. The
evaluated criteria is according to the CPU time and memory. In
this paper, two parts of the solution of the finite element
eigenvalue problem are presented, including the condensation
procedure and the direct eigenvalue extraction technique. The
condensation procedure used is based on Guyan reduction and the
idea of the substructure method is also based on Guyan
reduction. The available direct eigenvalue extraction procedure
includes the HQL(Householder-QL-inverse) method, generalized
Jacobi method,subspace method and Lanczos method. Each metthod
is discussed according to the speed of the CPU execution time
and the accuracy of the solution. Ls method, which using the
Lanczos method as the initial guess base vector in the subspace
iteration, has excellent performance according to the
computation speed and the accuracy of the solution. In last
part of this paper, machine tool modal analysis is used as an
example to study the efficiency of the substructure method
which is often used in large order eigenvalue problem.