First-order Maxwell Equations Research Articles

NEW EFFICIENT IMPLICIT TIME INTEGRATION METHOD FOR DGTD APPLIED TO SEQUENTIAL MULTIDOMAIN AND MULTISCALE PROBLEMS

The discontinuous Galerkin's (DG) method is an efficient technique for packaging problems. It divides an original computational region into several subdomains, i.e., splits a large linear system into several smaller and balanced matrices. Once the spatial discretization is solved, an optimal time integration method is necessary. For explicit time stepping schemes, the smallest edge length in the entire discretized domain determines the maximal time step interval allowed by the stability criterion, thus they require a large number of time steps for packaging problems. Implicit time stepping schemes are unconditionally stable, thus domains with small structures can use a large time step interval. However, this approach requires inversion of matrices which are generally not positive definite as in explicit shemes for the first-order Maxwell's equations and thus becomes costly to solve for large problems. This work presents an algorithm that exploits the sequential way in which the subdomains are usually placed for layered structures in packaging problems. Specifically, a reordering of interface and volume unknowns combined with a block LDU (Lower-Diagonal-Upper) decomposition allows improvements in terms of memory cost and time of execution, with respect to previous DGTD implementations.

Interior Penalty Discontinuous Galerkin Finite Element Method for the Time-Dependent First Order Maxwell's Equations

An interior penalty discontinuous Galerkin method is described with a conformal perfectly matched layer (PML) for solving the two first-order Maxwell's equations in the time domain. Both central and upwind fluxes are studied in this work. In both cases, the proposed method is explicit and conditionally stable. Additionally, a local time-stepping strategy is applied to increase efficiency and reduce the computational time. Finally, numerical examples are presented to validate the method.

A Nonspurious 3-D Vector Discontinuous Galerkin Finite-Element Time-Domain Method

We propose a nonspurious vector discontinuous Galerkin finite-element time-domain (DG-FETD) method for 3-D electromagnetic simulation. To facilitate the implementation of numerical fluxes for domain decomposition, we construct the DG-FETD scheme based on the first-order Maxwell's equations with variables E and H. The LT/QN and the CT/LN edge elements are employed to represent E and H, respectively (or vice versa), to suppress spurious modes, and the Riemann solver is utilized as the numerical flux to correct fields on the interfaces between adjacent subdomains. Numerical experiments show the nonspurious property of the proposed method.

1+1+2 electromagnetic perturbations on general LRS spacetimes: Regge–Wheeler and Bardeen–Press equations

We use the (covariant and gauge-invariant) 1+1+2 formalism developed by Clarkson and Barrett (2003 Class. Quanum Grav. 20 3855–84), and develop new techniques, to decouple electromagnetic (EM) perturbations on arbitrary locally rotationally symmetric (LRS) spacetimes. Ultimately, we derive three decoupled complex equations governing three complex scalars. One of these is a new Regge–Wheeler (RW) equation generalized for LRS spacetimes, whereas the remaining two are new generalizations of the Bardeen–Press (BP) equations. This is achieved by first using linear algebra techniques to rewrite the first-order Maxwell equations in a new complex 1+1+2 form which is conducive to decoupling. This new complex system immediately yields the generalized RW equation, and furthermore, we also derive a decoupled equation governing a newly defined complex EM 2-vector. Subsequently, a further decomposition of the 1+1+2 formalism into a 1+1+1+1 formalism is developed, allowing us to decompose the complex EM 2-vector, and its governing equations, into spin-weighted scalars, giving rise to the generalized BP equations.

High-Order Symplectic Integration Methods for Finite Element Solutions to Time Dependent Maxwell Equations

In this paper, we motivate the use of high-order integration methods for finite element solutions of the time dependent Maxwell equations.. In particular, we present a symplectic algorithm for the integration of the coupled first-order Maxwell equations for computing the time dependent electric and magnetic fields. Symplectic methods have the benefit a conserving total electromagnetic field energy and are, therefore, preferred over dissipative methods (such as traditional Runge-Kutta) in applications that require high-accuracy and energy conservation over long periods of time integration. We show that in the context or symplectic methods, several popular schemes can be elegantly cast in a single algorithm. We conclude with some numerical examples which demonstrate the superior performance of high-order time integration methods.

Analysis of Rotman lenses using a hybrid least squares FEM/transfinite element method

A hybrid procedure that combines the least squares finite element method (LSFEM) and the transfinite element method is applied to the analysis of multiport Rotman lenses. The LSFEM is employed to build a functional based on the L2-norm of the residuals of the full first-order Maxwell equations governing the electromagnetic fields in the structure. Because the full first-order Maxwell equations are solved, the method is free from spurious solutions. The transfinite element method is used to transfer the exterior conditions in the port regions to the interior region. The method is applied to a Rotman lens to determine the field distribution in the lens and the scattering parameters. Measured results for a microstrip Rotman lens with nine beam ports and eight array ports show good agreement with theoretical predictions.

The Origin of Spurious Solutions in Computational Electromagnetics

It is commonly believed that the divergence equations in the Maxwell equations are “redundant” for transient and timeharmonic problems, therefore most of the numerical methods in computational electromagnetics solve only two first-order curl equations or the second-order curl–curl equations. This misconception is the true origin of spurious modes and inaccurate solutions in computational electromagnetics. By studying the div–curl system this paper clarifies that the first-order Maxwell equations are not “overdetermined,” and the divergence equations must always be included to maintain the ellipticity of the system in the space domain, to guarantee the uniqueness of the solution and the accuracy of the numerical methods, and to eliminate the infinitely degenerate eigenvalue. This paper shows that the common derivation and usage of the second-order curl–curl equations are incorrect and that the solution of Helmholtz equations needs the divergence condition to be enforced on an associated part of the boundary. The div–curl method and the least-squares method introduced in this paper provide rigorous derivation of the equivalent second-order Maxwell equations and their boundary conditions. The node-based least-squares finite element method (LSFEM) is recommended for solving the first-order full Maxwell equations directly. Examples of the numerical solutions by LSFEM are given to demonstrate that the LSFEM is free of spurious solutions.

Frequency‐domain modelling of airborne electromagnetic responses using staggered finite differences1

AbstractA 3D frequency‐domain EM modelling code has been implemented for helicopter electromagnetic (HEM) simulations. A vector Helmholtz equation for the electric fields is employed to avoid convergence problems associated with the first‐order Maxwell's equations when air is present. Additional stability is introduced by formulating the problem in terms of the scattered electric fields. With this formulation the impressed dipole source is replaced with an equivalent source, which for the airborne configuration possesses a smoother spatial dependence and is easier to model. In order to compute this equivalent source, a primary field arising from dipole sources of either a whole space or a layered half‐space must be calculated at locations where the conductivity is different from that of the background.The Helmholtz equation is approximated using finite differences on a staggered grid. After finite‐differencing, a complex‐symmetric matrix system of equations is assembled and preconditioned using Jacobi scaling before it is solved using the quasi‐minimum residual (QMR) method. The modelling code has been compared with other 1D and 3D numerical models and is found to produce results in good agreement.We have used the solution to simulate novel HEM responses that are computationally intractable using integral equation (IE) solutions. These simulations include a 2D conductor residing at a fault contact with and without topography. Our simulations show that the quadrature response is a very good indicator of the faulted background, while the in‐phase response indicates the presence of the conductor. However when interpreting the in‐phase response, it is possible erroneously to infer a dipping conductor due to the contribution of the faulted background.