有限元素法之固有值問題在多邊形鼓膜上之誤差分析

The finite element method of the eigenvalue problem for the analysis of error with polygonal membrane

徐茂霖(Mau-Ling Hsu)

指導教授 ： 周謀鴻

國立臺灣大學/理學院/數學研究所/碩士(2006年)

https://doi.org/10.6342/NTU.2006.01258

摘要

無資料

關鍵字

有限元素法 ； 固有值

並列摘要

This Helmholtz equation occurs frequently in dynamic meteorology. In classical physics it is the equation of the vibrating membrane. Time-harmonic wave propagation, either elastic waves or electromagnetic waves, is a common phenomenon that appears in many applications such as acoustic wave scattering from submarines, noise reduction in silencers and mufflers, earthquake wave propagation.

First, the Helmholtz equation reduces to that of solving a generalized eigenvalue problem. In the chapter 2, it shows that the order of the error of eigenvalue is O(h^2) by the application of the spectral theory.

In the latest chapter, the Helmholtz equation can be solved via finite element methods. In these numerical results, the eigenvalues are solved throughout a linear triangular element is recognized as approximation. The eigenvalue are solved throughout a quadratic triangular element is recognized as exact solution, since this numerical method has much higher rate of convergence. In addition, we compare the difference between linear triangular element and quadratic triangular element by refining the mesh.

並列關鍵字

finite element method ； eigenvalue

參考文獻

[ 1 ] Kato,T.(1958), “ Perturbation theory for nullity. ” 261-322.

[ 2 ] Ciarlet, P.G. and Lions J.L.(1991), “ Handbook of numerical analysis. ” 683-699.

[ 3 ] Babuska, I. and A. Aziz (1973) “ Survey lectures on the mathematical foundations of the finite element method. ” 5-359

[ 4 ] William G. Kolata(1978) “ Approximation in variationally posed eigenvalue problems. ” Numer. Math. 29, 159-171

[ 5 ] Ciarlet, P.G.(2002) “ The finite element method for elliptic problem.”