An efficient and accurate hybrid spectral element-spectral integral (SESI) method is proposed to simulate electromagnetic (EM) scattering by objects in multiregion cylindrically layered media. The field in each of the multiple regions with multiple scatterers can be solved by the spectral element method (SEM), while the spectral integral method (SIM) is used as an exact boundary condition for the adjacent regions with multilayered media. This boundary condition between the electric field and the surface current density in each cylindrical layer can be obtained by using a recursive formula in SIM. The inverse matrix of a Toeplitz matrix arising from the SIM can be solved by the fast Fourier transform (FFT) with $O(N\log N)$ complexity. The hybrid SESI method is used to simulate homogeneous and/or inhomogeneous multiregion multilayered structures. Numerical examples, including the frequency selective surface (FSS) applications, demonstrate that the hybrid SESI method is efficient and accurate by comparing its results with analytical solutions and results from the conventional SEM and the finite element method (FEM).

A Hybrid SESI Method for Electromagnetic Scattering by Objects in Multiregion Cylindrically Layered Media

Combining the traditional Nitsche’s method (NM) and Gauss’ law, we propose a mixed Nitsche’s method (MNM) based on a hybrid tetrahedron–hexahedron mesh. With this method, geometrical models can be meshed flexibly and zero-frequency (dc) spurious modes can be effectively removed when solving Maxwell’s eigenvalue problems. The number of degrees of freedom (DOFs) in the MNM remains the same as that in the NM by using the tree–cotree technique. Moreover, we summarize the rules of selecting tree edges for high-order hierarchical basis functions and derive the constrained equations with discrete gradient matrices to suppress Ritz vectors in the Arnoldi algorithm. The MNM is not only suitable for a hybrid mesh but also for purely hexahedral and purely tetrahedral meshes, and there is no need to calculate the discrete gradient matrices and to modify the Arnoldi algorithm. Through numerical experiments, it is found that the MNM converges exponentially for a cube cavity discretized by a hybrid mesh and its efficiency is about four times higher than the commercial software COMSOL. 

Mixed Nitsche’s Method for Maxwell’s Eigenvalue Problems

We use the fully discrete interpolation coefficient mixed finite element methods to solve the semi-linear parabolic optimal control problems. The space discretization of the state variable is separated using interpolation coefficient mixed finite elements. We approximate the state and the co-state by Raviart-Thomas mixed finite elements on the lowest order, which is approximated by piecewise constant elements. By applying mixed finite element methods to estimate errors, we get priori errors estimates for both the coupled state and the control approximations. We finally confirm the theoretical results numerically by a numerical example.

Fully Discrete Interpolation Coefficients Mixed Finite Element Methods for Semi-Linear Parabolic Optimal Control Problem

In this paper, the symmetric Lanczos algorithm and mixed finite element method are combined to improve the efficiency and accuracy of S-parameter simulation in a broad frequency band. The mixed finite element method based on tree-cotree splitting is adopted to overcome the low-frequency (high-density) breakdown problem, thus ensuring the accuracy of low frequency (high integration density) results. The modified symmetric Lanczos algorithm for second order eigenvalue problem is employed for fast frequency sweep by one matrix inversion, then the field in the entire frequency band is obtained rapidly by solving the reduced order matrix equation. The accuracy of the high frequency results is improved by increasing the number of Lanczos vectors. Numerical simulations of two integrated passive filters (IPD filter) and an interconnect model are shown to verify the hybrid method along with commercial software HFSS. 

Broadband S-Parameter Simulation Based on the Mixed Finite-Element Method and Symmetric Lanczos Algorithm

Band Structure Calculations of Three-Dimensional Anisotropic Photonic Crystals in the Oblique Coordinate System

Two mixed spectral-element methods (MSEMs) are proposed to solve Maxwell’s eigenvalue problems with Bloch (Floquet) periodic and open resonators to remove the dc spurious modes present in the traditional numerical methods. The first MSEM combines the variational form of Gauss’s law to make the electric field satisfy the divergence-free condition. This method does not need to modify the Arnoldi algorithm for the eigenvalue problems, but the number of degrees of freedom (DOFs) is larger than that of the traditional spectral-element method (SEM). The second MSEM, as a generalization of the tree–cotree technique, can eliminate the dc spurious modes by imposing the constrained equations on the Ritz vectors in the Arnoldi algorithm without increasing the DOF from the traditional SEM. We expand the electric field in terms of the edge basis functions associated with the cotree edges and the gradient of the nodal basis functions so that the discrete gradient matrix $\bar {\bar {\mathrm {G}}}$ consists of the identity matrix $\bar {\bar {\mathrm {G}}}_{t}$ and the zero matrix $\bar {\bar {\mathrm {G}}}_{c}$ , making the matrix inversion of $\bar {\bar {\mathrm {G}}}_{t}$ trivial. Finally, in the numerical experiments, we compare the computational costs of the MSEMs and finite-element method (FEM) in COMSOL to illustrate the high efficiency of the MSEMs.

Mixed Spectral-Element Methods for 3-D Maxwell’s Eigenvalue Problems With Bloch Periodic and Open Resonators

In order to simulate elastic wave propagation in a complex structure with inhomogeneous media, we often need to obtain the propagating eigenmodes of an elastic waveguide. As the waveguide is assumed uniform in one direction, the original 3-D problem can be converted into a so-called 2.5-D problem by using the Fourier transform in that direction. However, the introduction of elastic metamaterials (EMM) broadens the horizon of this subject, and new features are required in EMM waveguides that cannot be obtained by most traditional waveguide solvers. In this work, a spectral element method (SEM) is developed to simulate the elastic/acoustic waveguide problem in anisotropic media with anisotropic mass density and/or negative index parameters. To the best of our knowledge, the SEM has not been introduced previously for such a waveguide problem. For waveguides with anisotropic density that cannot be solved by the FEM in most of commercial software packages, we design an anisotropic density EMM waveguide with our SEM solver to demonstrate some intriguing phenomena. The spectral element results are verified by several numerical examples through comparison with the traditional finite element method (FEM) to show its significant advantages in term of accuracy and computation efficiency.

Ming Wei Zhuang

Papers

Mixed Spectral-Element Methods for 3-D Maxwell’s Eigenvalue Problems With Bloch Periodic and Open Resonators

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