由表面聲波濾通所產生之辛特徵問題的保結構數值法

Abstract

週期表面聲波過濾(SAW-filters)的數值模擬及色散圖樣的計算在電子通訊工業中扮演了重要的角色。此類物理現象由波動方程來描述，在時間harmonic及空間quasi-harmonic的Bloch-Flquent定理推導下，我們得到了一Helmholtz型態的特徵值問題。利用了Finite-element方法配合適當的邊界條件，我們可以導出二種不同的特徵值問題：(1) Palindromic 二次特徵值；(2) 辛對特徵值問題。每一問題皆擁有(λλ1,)的辛對譜。本計劃主旨是要發展一套有效的保結構計算方法，對傳統軟體所不能求得高精度之辛對譜作出高精度的修正。

In telecommunication industry, computation of so-called “dispersion diagram” and numerical simulation of periodic surface acoustic wave filters (SAW-filters) play an important role. The principle of surface acoustic wave filter is based on the physical properties of piezoelectric materials. The main components of a SAW filter are a piezoelectric substrate and an input and an output inter-digital transducer (IDT). The propagation of surface wave is characterized by the frequency of input signal, by the geometrical arrangements of the electrodes, and by the material parameters of the piezoelectric substrate and electrodes. Due to the underlying geometry no surface wave can propagate at special frequency intervals. Therefore, these frequencies are missing in the output signal and the device can be used for frequency filtering. The frequency domain can be parted into pass-bands (i.e. frequencies which get through the piezoelectric device) and stop-bands (i.e. frequencies which get filtered out). For given geometry and material parameters propagation and attenuation of acoustic waves and frequency can be thought as so-called “dispersion diagram.” In this project, we focus on SAW-filters in wireless communication, e.g., standard components in TV-sets and cellular phone. Therefore, we assume that excitation force is time-harmonic (with /2fωπ=) and quasi-periodic in space, i.e., ()(,)()ixitpfxyfxeeαβω+=, where ()pfx is a p-periodic function. By Bloch-Floquet theory the solution on periodic structures can be decomposed in quasi-periodic functions so-called Bloch waves. Therefore, the problem can be restricted to the unit-cell (including one electrode). Such an one-cell problem turns out a Helmholtz-type problem. In numerical computation we use finite element method with appropriate boundary conditions to establish two different type of eigenvalue problems: (1) palindromic quadratic eigenvalue problem, (2) symplectic eigenvalue problem. Each problem has eigenvalue λ and into reciprocal 1/λ. Unfortunately classical FEM packages produce answers that fail to deliver even a single correct digit. In this project, we would like to develop a structured backward stable algorithm by use of -transform to solve palindromic QEP. And then to develop a generalized shift-invert implicitly restarted Arnoldi-type algorithm for solving large sparse palindromc QEP. The whole ideal here is careful exploitation of the symplectic structure of the eigenvalue problem.