彈性波內部傳輸特徵值問題的計算

Abstract

這項工作旨在研究在非均勻介質中彈性波散射的內部傳輸特徵值問題。根據散射域中是否嵌入障礙物，我們可以建立兩種偏微分方程，再使用有限元方法分別導出各別的廣義特徵值問題。在具有內嵌障礙物的問題中，根據內嵌障礙物的邊界條件，我們可以考慮更多因素。我們旨在計算大量且最接近目標的傳輸特徵值。在每種特徵值問題中皆考慮兩種參數變化，即跳躍彈性和跳躍密度。基於Jacobi-Davidson 方法，我們提供了一些可以有效估計所需特徵值的算法。特別是，對於跳躍密度，我們必須在算法開始時另外刪除非物理零特徵值，其數量取決於離散化後點的個數。針對沒有內嵌障礙物的情況，為了解決這個難題，我們特別將廣義特徵值問題轉換為二次特徵值問題。相對地，對於內嵌障礙物的問題，我們將零特徵值轉換到無窮大。本論文將介紹算法並給出其數值結果。

This work aims to study the interior transmission eigenvalue problem (ITEP) for elastic waves scattering in inhomogeneous media. Depending on whether there is an obstacle embedded in the scattering domain, we can establish two partial differential equations and separately derive the generalized eigenvalue problem (GEP) using the finite element method (FEM). In a problem with an embedded obstacle, we can consider more factors because of the boundary condition of the obstacle. We aim to compute a large number of transmission eigenvalues that are closest to the target. Two parameters, i.e., jumping elasticity and jumping density, are considered in each eigenvalue problem. Based on the Jacobi-Davidson (JD) method, we provide some algorithms that can efficiently estimate the desired eigenvalues. In particular, for jumping density, we must additionally remove the nonphysical zero eigenvalues whose number depends on the size of the discretization at the beginning of the algorithms. To deal with this difficulty, we transform the GEP into a quadratic eigenvalue problem (QEP) for a problem without an embedded obstacle. Moreover, for a problem with an embedded obstacle, we deflate the zero eigenvalues to infinity. In this dissertation, we introduce the algorithms and present their numerical results.