多邊形鼓膜振動問題固有值之數值分析

Abstract

When the drumhead vibrating, there are some lines keep still. These “nodal lines” are what we are interesting in. In this text, we want to solve the eigenvalue problem on right polygonal area and find out some properties about the eigenvalues from the numerical results.

First, we divide the drumhead into triangle elements and use the finite element method to generate a matrix generalized eigenvalue problem. Second, apply the Cholesky decomposition on the so-called “mass matrix” and transfer the problem to a standard eigenvalue problem. Finally, we use the Householder reflections to simplify the matrix we want to solve into tridiagonal form and apply shifted QR algorithm to get its approximation eigenvalues.

In the latest chapter, we will make some observations from the numerical results. By varying the drumhead’s edge number, area and element’s size, we can see the different changes about the eigenvalues. But when the drumhead’s edge number increase, we can find that both the eigenvalues and nodal sets will converge.