Mixed Spectral Element Method Research Articles

Pure TE and TM modes in the metallic waveguide filled homogeneously with fully anisotropic and lossless medium

AbstractThis letter gives a sufficient condition for the existence of the pure transverse electric (TE) and transverse magnetic (TM) modes in the metallic waveguide filled with a homogeneous, fully anisotropic and lossless medium. The condition is a relation between the permittivity and permeability tensors of the fully anisotropic medium. The theory given in the letter is an extension of the classic electromagnetic waveguide theory. Furthermore, a mixed spectral element method is applied to simulate this type of anisotropic waveguide problem. At last, a numerical experiment shows that the sufficient condition supported in the letter does confirm the existence of the pure TE and TM modes in the anisotropic waveguide.

Advanced Numerical Methods for Graphene Simulation with Equivalent Boundary Conditions: A Review

Since the discovery of graphene, due to its excellent optical, thermal, mechanical and electrical properties, it has a broad application prospect in energy, materials, biomedicine, electromagnetism and other fields. A great quantity of researches on the physical mechanism of graphene has been applied to engineering in electromagnetism and optics. To study the properties of graphene, different kinds of numerical methods such as the mixed finite element method (Mixed FEM), the mixed spectral element method (Mixed SEM), Method of Auxiliary Sources (MAS), discontinuous Galerkin time-domain method (DGTD) and interior penalty discontinuous Galerkin time domain (IPDG) have been developed for simulating the electromagnetic field effects of graphene and equivalent boundary conditions such as impedance transmission boundary condition (ITBC), surface current boundary condition (SCBC), impedance matrix boundary condition (IMBC) and surface impedance boundary condition (SIBC) have been employed to replace graphene in the computational domain. In this work, the numerical methods with equivalent boundary conditions are reviewed, and some examples are provided to illustrate their applicability.

3-D NMM Method for Fully Anisotropic and Nonreciprocal Media

A new 3-D spectral numerical mode-matching (SNMM) method is developed to simulate electromagnetic waves in multilayered structures with fully anisotropic and nonreciprocal media. The SNMM method reduces the original 3-D layered structure into a series of 2-D waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with fully anisotropic and nonreciprocal media are solved numerically by the mixed spectral element method (MSEM) with exponential convergence. Based on the waveguide eigenmodes, the local reflection and transmission matrices and the global reflection matrices are expanded so that the new NMM method is suitable for fully anisotropic and nonreciprocal structures. The NMM method is applied to solve three models, including a three-layer symmetrical structure with inhomogeneous nonreciprocal media, the negative refraction of a YVO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> bicrystal, and a 3-D multilayer structure with nonreciprocal media. Numerical experiments indicate that this new NMM method is highly efficient and accurate for these models for fully anisotropic and nonreciprocal media.

3-D Numerical Mode Matching Method for Off- Centered Electromagnetic Well Logging Tools in Noncircular Vertical Borehole and Invasion Zones in Multilayered Media

In electromagnetic (EM) well logging for petroleum exploration, off-centered tools within a noncircular borehole and invasion zones in multilayered media represent a challenging 3-D problem for traditional numerical methods. The 3-D finite-element numerical mode-matching (FNMM) method and the spectral numerical mode-matching (SNMM) method are developed to address this problem in this work. The numerical mode-matching (NMM) methods reduce the original 3-D well logging problem into a series of 2-D open waveguide eigenvalue problems plus a 1-D layered medium problem, which can be analytically solved by a recursion procedure. These waveguide eigenvalue problems with anisotropic inhomogeneous media in the horizontal directions are solved numerically by the mixed finite-element method (MFEM) with the flexibility for the complex geometry and the mixed spectral element method (MSEM) with the exponential convergence. A general operator form of Maxwell’s equations is also derived to obtain the source excitation vector independent of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> -axis. Therefore, the NMM solvers can effectively and accurately simulate the response of off-centered EM well logging tools in an irregular borehole in complex formations. The NMM methods are applied to simulate three formations including a five-layer isotropic homogeneous formation, an off-centered logging tool in a noncircular borehole with anisotropic invasion zones in a five-layer anisotropic formation, and the Oklahoma formation. Numerical experiments indicate that the NMM methods are highly efficient and accurate for the EM well logging models.

Mixed Spectral Element Method for Electromagnetic Secondary Fields in Stratified Inhomogeneous Anisotropic Media

The spectral element method (SEM) is widely used for analyzing high frequency electromagnetic wave propagation and scattering in complex structures. At low frequencies, however, the primary (direct) fields are usually much larger than the secondary fields caused by scattering, thus making it much more difficult to accurately simulate the secondary (scattered) fields because of the source singularity in the primary fields. To more effectively and accurately simulate the low-frequency secondary fields in subsurface structures, in this work the vector Helmholtz equation for the scattered fields is used to directly obtain the secondary field data in inhomogeneous anisotropic media. The computational method for the secondary electromagnetic fields is based on the mixed spectral element method (mixed SEM) for anisotropic objects in a layered anisotropic background medium. The electromagnetic field is separated into the background fields that are evaluated analytically using the dyadic Green's functions (DGFs) of a layered uniaxially anisotropic medium, and the secondary fields caused by anomalous bodies with arbitrary shapes are numerically computed by the mixed SEM. This avoids the source singularity, and allows the transmitters to be located outside the computational domain. By enforcing Gauss' law as the constraint condition, the mixed SEM efficiently eliminates the low frequency breakdown problem. Numerical results verify the proposed method over the traditional SEM in low-frequency scattering from objects in subsurface layered anisotropic media.

Modal Analysis of 2-D Material-Based Plasmonic Waveguides by Mixed Spectral Element Method With Equivalent Boundary Condition

Recently, atomically thin two-dimensional (2-D) materials have attracted significant attention due to their promising applications in optoelectronic devices. Among them, surface plasmon-polaritons (SPPs) stimulated by 2-D material-based waveguide provide a new mechanism to propagate and enhance lightwaves in the deep subwavelength. However, the modeling and simulation of the 2-D material-based structures are still a challenge due to the mismatch between their electrically small in thickness and electrically large in transversal cross-section. In order to obtain a high-efficiency modal analysis of 2-D material-based plasmonic waveguides, the mixed spectral element method (mixed-SEM) with surface current boundary condition (SCBC) is proposed in this paper. The new method introduces a variational formulation that combines the transversal vectorial Helmholtz equation with the Gauss' law, and a SCBC that is implemented as the equivalent boundary condition to eliminate 2-D nanoscale thin sheets from the computational domain. It thus avoids an enormous number of unknowns and significantly reduces the computational costs. To verify the proposed method, some typical 2-D material-based plasmonic waveguides including those composed of anisotropic black phosphorus are analyzed and numerical results are compared in various ways. Excellent agreements are obtained for all the cases which clearly suggest that the proposed method can efficiently and accurately simulate 2-D materials-based plasmonic waveguides.

The Efficient Hybrid Mixed Spectral Element Method With Surface Current Boundary Condition for Modeling 2.5-D Fractures and Faults

Fast electromagnetic (EM) simulation of thin layers, such as geological fractures and faults, is still a challenge since the local mesh refinement near such structures dramatically increases the number of degrees of freedom (DoFs), thus large computational resources are required for modeling their interactions with electromagnetic fields. Therefore, significant attention has been paid to overcome the problem. In this work, we propose the 2.5 dimensional hybrid mixed spectral element method with a surface current boundary condition (the 2.5-D hybrid MSEM-SCBC) to accelerate the simulation and utilize it for the low-frequency solutions of electromagnetic field in subsurface sensing. By introducing SCBC to model the EM field diffraction, the geological fractures and faults are represented geometrically by zero-thickness sheets instead of finite thickness thin layers to avoid local mesh refinement, and to significantly decrease DoFs and computational costs. Numerical results demonstrate that the proposed method is efficient and accurate in simulation of 2.5-D subsurface sensing problems.

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

The mixed spectral-element method (MSEM) is applied to solve the waveguide problem with the 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 spectral-element method (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 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 Metasurfaces

A novel 3-D spectral numerical mode-matching (SNMM) method is proposed and developed as a highly efficient rigorous solver for metasurfaces on the interface of a half-space with Bloch (Floquet) periodic boundary conditions. In addition to full electromagnetic (EM) fields, the SNMM solver can provide with ease the characteristics of metasurfaces including their absorptance, anomalous reflection/refraction, and surface plasmon polaritons (SPPs). The SNMM method is a semianalytical solver: it solves for the Bloch eigenmodes in the horizontal directions by using the mixed spectral-element method (MSEM) numerically, but determines the scattering in the vertical direction analytically through eigenmode propagation. As there is no need for discretization in the vertical direction, it can efficiently and accurately simulate EM wave interactions with metasurfaces. Numerical experiments indicate that the SNMM method is efficient and accurate for the metasurfaces compared with the well-developed 3-D finite-element method (FEM). Applications to homogeneous isotropic/anisotropic, inhomogeneous isotropic, and the gradient metasurfaces are demonstrated. Typically, the computational speed of the proposed solver is one to two orders of magnitude higher than the FEM as implemented in a commercial software package.

Spectral-Element Method With Divergence-Free Constraint for 2.5-D Marine CSEM Hydrocarbon Exploration

Rapid simulations of large-scale low-frequency subsurface electromagnetic measurements are still a challenge because of the low-frequency breakdown phenomenon that makes the system matrix extremely poor-conditioned. Hence, significant attention has been paid to accelerate the numerical algorithms for Maxwell’s equations in both integral and partial differential forms. In this letter, we develop a novel 2.5-D method to overcome the low-frequency breakdown problem by using the mixed spectral element method with the divergence-free constraint and apply it to solve the marine-controlled-source electromagnetic systems. By imposing the divergence-free constraint, the proposed method considers the law of conservation of charges, unlike the conventional governing equation for these problems. Therefore, at low frequencies, the Gauss law guarantees the stability of the solution, and we can obtain a well-conditioned system matrix even as the frequency approaches zero. Several numerical experiments show that the proposed method is well suited for solving low-frequency electromagnetic problems.

Mixed Spectral-Element Method for Overcoming the Low-Frequency Breakdown Problem in Subsurface EM Exploration

One fundamental difficulty in low-frequency subsurface electromagnetic exploration is the low-frequency breakdown phenomenon in numerical computation. It makes the discretized linear system very poorly conditioned and thus difficult to solve. This issue is present in both integral equation and partial differential equation solution methods, and thus has attracted many researchers who have proposed various methods to overcome this difficulty. In this paper, we propose a new mixed spectral element method (mixed SEM) to eliminate this low-frequency breakdown problem and apply this method to solve the subsurface electromagnetic exploration problem. Since Gauss’ law is now explicitly enforced in the mixed SEM to make the system matrix well-conditioned even at extremely low frequency, we can solve the linear system from dc to high frequencies. With the proposed method, we study the surface-to-borehole electromagnetic system for hydrocarbon exploration. Numerical examples show that the mixed SEM is accurate and efficient, and has significant advantages over conventional methods.

Mixed Spectral-Element Method for 3-D Maxwell's Eigenvalue Problem

A high-order 3-D mixed spectral-element method (SEM) based on Gauss-Lobatto-Legendre polynomials in the mixed finite-element framework is proposed to remove the spurious eigenmodes in the numerical solution of the vector Maxwell eigenvalue problem with inhomogeneous, lossy isotropic, and anisotropic media. In order to suppress all zero and nonzero spurious modes that exist in the conventional finite-element and higher order SEMs, the proposed method not only employs the mixed-order curl-conforming vector basis functions for the electric field intensity, but also includes the divergence-free condition given by Gauss' law in a weak form. Several numerical examples are given to verify that the mixed SEM is free of any spurious eigenmodes and has spectral accuracy with analytic eigenvectors.

Mixed Spectral Element Method for 2D Maxwell's Eigenvalue Problem

AbstractIt is well known that conventional edge elements in solving vector Maxwell's eigenvalue equations by the finite element method will lead to the presence of spurious zero eigenvalues. This problem has been addressed for the first order edge element by Kikuchi by the mixed element method. Inspired by this approach, this paper describes a higher order mixed spectral element method (mixed SEM) for the computation of two-dimensional vector eigenvalue problem of Maxwell's equations. It utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element framework with a weak divergence condition. It is shown that this method can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous analysis of the convergence of the mixed SEM is presented, based on the higher order edge element interpolation error estimates, which fully confirms the robustness of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape, coaxial and dual-inner-conductor cavities to verify the merits of the proposed method.

Staggered grids hybrid-dual spectral element method for second-order elliptic problems application to high-order time splitting methods for Navier-Stokes equations

This paper is a presentation of a mixed spectral element method, based on staggered grids strategy, used for the discretization of second-order elliptic problems set under hybrid dual formulation. A numerical analysis is performed proving the expected optimality and some numerical tests are described to confirm the predicted convergence rates. The second part deals with a high-order time splitting/spectral scheme for the simulation of unsteady incompressible Navier-Stokes flows. We show how to use such a solver as a filtering tool to obtain divergence free discrete velocity. Some computational examples are reported which quantify the improvements, induced by such a filter, we have observed: a strengthening of the stability and of the accuracy.