Mixed Spectral-Element Method for the Waveguide Problem With Bloch Periodic Boundary Conditions

Abstract

The mixed spectral element method (MSEM) is applied to solve the waveguide problem with Bloch periodic boundary condition (BPBC). Based on the BPBC for the original Helmholtz equation and the periodic boundary condition (PBC) for the equivalent but modified Helmholtz equation, two equivalent mixed variational formulations are applied for the MSEM. Unlike the traditional finite element method and SEM, both these mixed SEM schemes are completely free of spurious modes because of their use of the Gauss' law and the curl-conforming vector basis functions structured by the Gauss-Legendre-Lobatto (GLL) points. A simple implementation method is used to deal with the BPBC and the PBC for the mixed variational formulations so that both schemes can save computational costs over the traditional methods. Several numerical results are also provided to verify that both schemes are free of spurious modes and have high accuracy with the propagation constants.

Spectral Numerical Mode-Matching Method for 3-D Layered Multiregion Structures

“…As explained in [25], the eigenfunctions e t and e new z are further written as the plane wave forms…”

“…Along the way of [25], the eigenpair (u, w, k z ) can be obtained by using the MSEM. It is easy to see that once the eigenpair (u, w, k z ) is obtained from (6), then the eigenpair (e t , e new z , k z ) can be also obtained from (4).…”

“…where ξ is the component of phase gradient parallel to the incident plane, a 1 is the unit cell length in the x direction, and d denotes the thickness of the MM slab. From (25) and the identity M (x) = −jσ es /ω 0 d, we can derive the surface conductivityσ es (x) = j(k 0 d + 0.5ξx)/(120π) for this MM slab. As shown in Fig.…”

“…In this paper, a 3-D semi-analytical solver is developed to model the multi-region structure with doubly periodic boundary conditions in the horizontal directions, based on the NMM idea with the mixed spectral element method for the 2-D eigenvalue problem, hence it is called the spectral numerical mode-matching (SNMM) method. In order to obtain these high accuracy physical eigenmodes, the mixed spectral element method (MSEM) is employed to solve the Bloch periodic waveguide eigenvalue problems [25]. The MSEM is based on the spectral element method (SEM) and Gauss' law, which can remove all the nonphysical eigenmodes and achieve exponential convergence.…”

Spectral Element Method for the Elastic/Acoustic Waveguide Problem in Anisotropic Metamaterials

“…As an extension of the electromagnetic and acoustic waveguides 5,6 , Lagasse proposes a finite element method (FEM) for computing the eigenmodes of the homogeneous elastic waveguides of arbitrary cross sections 7 ; Kosmodamianskii et. al.…”