Browsing by Author "Chern, Ruey-Lin"

Now showing 1 - 2 of 2

Electromagnetic field behavior of 3D Maxwell's equations for chiral media
(2019-02-15)
Huang, Tsung-Ming
;
Li, Tiexiang
;
Chern, Ruey-Lin
;
Lin, Wen-Wei
;
應用數學系
This article focuses on numerically studying the eigenstructure behavior of generalized eigenvalue problems (GEPs) arising in three dimensional (3D) source-free Maxwell's equations with magnetoelectric coupling effects which model 3D reciprocal chiral media. It is challenging to solve such a large-scale GEP efficiently. We combine the null-space free method with the inexact shift-invert residual Arnoldi method and MINRES linear solver to solve the GEP with a matrix dimension as large as 5,308,416. The eigenstructure is heavily determined by the chirality parameter gamma. We show that all the eigenvalues are real and finite for a small chirality gamma. For a critical value gamma = gamma*, the GEP has 2 x 2 Jordan blocks at infinity eigenvalues. Numerical results demonstrate that when gamma increases from gamma*, the 2 x 2 Jordan block will first split into a complex conjugate eigenpair, then rapidly collide with the real axis and bifurcate into positive (resonance) and negative eigenvalues with modulus smaller than the other existing positive eigenvalues. The resonance band also exhibits an anticrossing interaction. Moreover, the electric and magnetic fields of the resonance modes are localized inside the structure, with only a slight amount of field leaking into the background (dielectric) material. (C) 2018 Elsevier Inc. All rights reserved.
SINGULAR VALUE DECOMPOSITIONS FOR SINGLE-CURL OPERATORS IN THREE-DIMENSIONAL MAXWELL'S EQUATIONS FOR COMPLEX MEDIA
(2015-01-01)
Chern, Ruey-Lin
;
Hsieh, Han-En
;
Huang, Tsung-Ming
;
Lin, Wen-Wei
;
Wang, Weichung
This article focuses on solving the generalized eigenvalue problems (GEP) arising in the source-free Maxwell equation with magnetoelectric coupling effects that models three-dimensional complex media. The goal is to compute the smallest positive eigenvalues, and the main challenge is that the coefficient matrix in the discrete Maxwell equation is indefinite and degenerate. To overcome this difficulty, we derive a singular value decomposition (SVD) of the discrete single-curl operator and then explicitly express the basis of the invariant subspace corresponding to the nonzero eigenvalues of the GEP. Consequently, we reduce the GEP to a null space free standard eigenvalue problem (NFSEP) that contains only the nonzero (complex) eigenvalues of the GEP and can be solved by the shift-and-invert Arnoldi method without being disturbed by the null space. Furthermore, the basis of the eigendecomposition is chosen carefully so that we can apply fast Fourier transformation (FFT-) based matrix vector multiplication to solve the embedded linear systems efficiently by an iterative method. For chiral and pseudochiral complex media, which are of great interest in magnetoelectric applications, the NFSEP can be further transformed to a null space free GEP whose coefficient matrices are Hermitian and Hermitian positive definite (HHPD-NFGEP). This HHPD-NFGEP can be solved by using the invert Lanczos method without shifting. Furthermore, the embedded linear system can be solved efficiently by using the conjugate gradient method without preconditioning and the FFT- based matrix vector multiplications. Numerical results are presented to demonstrate the efficiency of the proposed methods.