Time-harmonic Electromagnetic Fields Research Articles

Engaging Students in Applied Electromagnetics at the University of San Diego

Two possible topical approaches that have been applied to teaching an upper-division undergraduate electrical engineering applied electromagnetics course are presented. Each approach was applied to one of two offerings of the course, taught in different semesters. In either case, the course includes the study of electromagnetic theory and practical applications through laboratory experiments, design and projects. Using basic theory in time-harmonic electromagnetic fields developed early in the course, two different applications of the theory are emphasized in two alternate semesters of the course offering. The two applications that incorporated the common theoretical background of the course are: (1) applications in radio frequency engineering; and (2) radar and antenna design. An extension of the course to a topic beyond the confines of the course textbooks is included. The success of the course is further realized in two industry-sponsored capstone senior design projects.

Second-order statistics of complex observables in fully stochastic electromagnetic interactions: Applications to EMC

[1] The usefulness of stochastic methods to efficiently quantify uncertainties in computational models of electromagnetic interactions is illustrated. A refined study of the second-order moments of a complex-valued Thevenin model, which represents the coupling between a wire structure and a time-harmonic electromagnetic field, is presented. The configuration of a stochastically undulating thin wire illuminated by a stochastic incident plane wave is investigated in detail. Three computational methods are used to evaluate the mean values and covariance coefficients of the observable: a straightforward Cartesian-product quadrature method, a Monte-Carlo method, and a space-filling-curve method. The underlying patterns of the randomness are revealed by analyzing the covariance matrix' principal components. The study of this interaction configuration shows some general characteristics, which are expected to show up in any stochastic electromagnetic interaction problem. In particular, the results indicate that fluctuations in self-interaction coefficients (impedances) have distinct features and are quite different from the coefficients describing the interaction with externally generated fields (voltages).

A Coupled Mixed Finite Element Method for the Interaction Problem between an Electromagnetic Field and an Elastic Body

This paper deals with the coupled problem arising from the interaction of a time-harmonic electromagnetic field with a three-dimensional elastic body. More precisely, we consider a suitable transmission problem holding between the solid and a sufficiently large annular region surrounding it, and aim to compute both the magnetic component of the scattered wave and the stresses that take place in the obstacle. To this end, we assume Voigt's model, which allows interaction only through the boundary of the body, and employ a dual-mixed variational formulation in the solid medium. As a consequence, one of the two transmission conditions becomes essential, whence it is enforced weakly through the introduction of a Lagrange multiplier. An abstract framework developed recently, which is based on regular decompositions of the spaces involved, is applied next to show that our coupled variational formulation is well-posed. In addition, we define the corresponding Galerkin scheme by using PEERS in the solid and using the edge finite elements of Nédélec in the electromagnetic region. Then, we prove that the resulting coupled mixed finite element scheme is uniquely solvable and convergent. Moreover, optimal a priori error estimates are derived in the usual way. Finally, some numerical results illustrating the analysis and the good performance of the method are also reported.

Algorithmic Enhancements, Model-Order Reduction, and Multigrid Aspects in Contemporary Finite Element Implementations

This paper deals with time-harmonic electromagnetic fields only. It presents finite-element formulations of improved robustness and low-frequency behavior, fast <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">hp</i> multi-level solution techniques, and order-reduction methods that handle parameter-dependent systems at low computational cost.

Gumerovの2スカラポテンシャル法による多球モデル内時間調和電磁界解析

Time harmonic electromagnetic fields in multi-spheres models were calculated by the method of two scalar potentials proposed by Gumerov. The analyzed models were an eccentric two-spheres model and a ten-spheres model including brain core and a pair of eyeballs. The applied fields were a plane wave and a dipole magnetic field, and the frequency was 1.8 or 2.0GHz. The truncation degree p of multipole and local expansion was set to up to 170∼200. The peak power density on the x-y plane and the absorbed power in each tissue almost monotonously converged as p increased. It demonstrated the usefulness of the Gumerov's method.

Reciprocity gap music imaging for an inverse scattering problem in two-layered media

In this work we consider a modified MUSIC method for determining the positions of a collection of small perfectly conducting buried objects from measurements of time-harmonic electromagnetic fields on the surface of ground. This method is based on an asymptotic analysis of certain integrals of electric and magnetic fields, so-called reciprocity gap functionals, as the buried objects shrink to points. Unlike standard MUSIC reconstruction methods our algorithm avoids the computation of the Green's function for the background medium during the reconstruction process, but on the other hand it requires more measurement data. After describing the theoretical foundation of this reconstruction method, we provide numerical results showing its performance. We also compare these results to reconstructions obtained by a standard MUSIC algorithm.

Exact Gaussian Beam Expansion of Time-Harmonic Electromagnetic Waves

The present contribution is concern with an exact frame-based expansion of planar initial time-harmonic electromagnetic fields. The propagating field is described as a discrete superposition of tilted and shifted electromagnetic beam waveobjects over the frame spatial-spectral lattice. Explicit asymptotic expressions for the electromagnetic Gaussian beam propagators are obtained for the commonly used Gaussian windows.

Relaxed Incomplete Cholesky Factorization Preconditioned Symmetric LBCG Algorithm for Solving Electromagnetic Problems

The symmetric linear biconjugate gradient (S-LBCG) iterative algorithm combined with modified relaxed incomplete Cholesky (RIC) preconditioner is proposed for solving large sparse complex symmetric and highly indefinite systems, which results from the use of edge-based finite-element method (FEM) on the analysis of electromagnetic problems. The S-LBCG iterative algorithm is derived from the linear biconjugate gradient iterative algorithm. It contains only one matrix-vector product per iteration for solving complex sparse symmetric systems. The preconditioner is derived from the relaxed incomplete Cholesky factorization with diagonal compensation. More robust and efficient RIC preconditioner can be achieved by employing preprocessing scheme of scaling and reordering before the incomplete factorization. Numerical tests on analysis of time-harmonic electromagnetic field problems show that this RIC preconditioned S-LBCG algorithm performs excellently both on reducing the memory requirement and on convergence rate.

Electromagnetically induced localized ignition in secondary high explosives

A model for electromagnetically induced hot spots is developed from the well-established theories of dielectric mixtures, microwave absorption, heat transfer, and thermal ignition. This mathematical model is used to elucidate the interplay among these theories for a microwave heated system of secondary high explosive within which isolated electromagnetically lossy spheres are randomly distributed. Results are shown in this article for the specific case of pentaerythritol tetranitrate with embedded spheres of varying diameter and conductivity illuminated by a uniform time harmonic electromagnetic field of 1, 8, and 15 GHz. It is shown that for a given frequency and electric field strength internal to a particle embedded in a secondary high explosive, there exists a range of values for the particle’s diameter and conductivity for which electromagnetically induced hot-spot ignition of the high explosive is possible. By providing an accurate estimate for the range of necessary geometric and electrical properties of the system, this idealized model can be used to guide potentially complex and otherwise less tractable computational and experimental studies of microwave-induced hot spots. We found that the condition of relatively large particles, with semiconductorlike conductivity, exposed to the highest possible frequency is most favorable for hot-spot ignition.

An Asymptotic Factorization Method for Inverse Electromagnetic Scattering in Layered Media

We consider the inverse problem to reconstruct the number and the positions of a collection of finitely many small perfectly conducting scatterers buried within the lower halfspace of an unbounded two-layered background medium from near field measurements of time harmonic electromagnetic fields. For this purpose we first study the direct scattering problem and derive an asymptotic expansion of the scattered field, as the size of the scatterers tends to zero. Integral equation methods and a factorization of the corresponding near field measurement operator are applied to prove this expansion. In the second part of this work we use the asymptotic expansion to justify a noniterative reconstruction algorithm, which is a combination of factorization methods and MUSIC-type methods. We illustrate the feasibility of this method by a numerical example.

Determination of Frequency-Dependent Characteristics of Substation Grounding Systems by Vector Finite-Element Analysis

A three-dimensional finite-element tool was developed to compute time-harmonic electromagnetic fields and impedance of substation grounding systems. The formulation employs edge-based finite elements for the magnetic vector potential A and nodal shape functions for the electric scalar potential V. The method has been applied in several configurations presented in the literature. The results are compared with both analytical and experimental data reported by other authors, with overall good agreement

Robust GMRES recursive method for fast finite element analysis of 3D electromagnetic problems

AbstractA robust generalized minimal residual recursive (GMRESR) iterative method is proposed to solve a large system of linear equations resulting from the use of an un‐gauged vector‐potential formulation of finite element method (FEM). This method involves an outer generalized conjugate residual (GCR) method and an inner generalized minimal residual (GMRES) method, where the inner GMRES acts as a variable preconditioning for the outer GCR. The efficient implementation of symmetric successive overrelaxation (SSOR) preconditioned GMRESR (SSOR‐GMRESR) algorithm is described in details for complex coefficient matrix equation. On several three‐dimensional electromagnetic problems, the resulting SSOR‐GMRESR approach converges in CPU time, which is 14.2–71.3 times shorter with respect to conventional conjugate gradient (CG) approach. By comparison with other popularly preconditioned CG methods, the results demonstrate that SSOR‐GMRESR is especially effective and robust when the A‐V formulation of FEM is applied to solve large‐scale time harmonic electromagnetic field problems. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 1010–1015, 2007; Published online in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/mop.22333

Preconditioned iterative solver on the coarsest level of a multi-grid method for high frequency time harmonic electromagnetic field analyses

A multi-grid method is one of the most powerful linear solvers for finite element electromagnetic field analysis. However, as the discretized model has recently been enlarged, a solution process for a linear system arising on the coarsest level tends to be problematic in a complete multi-grid solution process. Whereas a linear system on the coarsest level is generally solved by a direct solver, we solve it here by means of an iterative solver to reduce the memory requirements. Since a conventional preconditioning technique is not effective for such a linear system, we introduce preconditioning techniques based on Arnold, Falk, and Winther’s and on Hiptmair’s smoothers. Numerical tests show that the newly installed preconditioning technique greatly improves the convergence rate.

Two-Tone Response of Radiofrequency Signals Using the Voltage Output of a Superconducting Quantum Interference Filter

In the presence of weak time-harmonic electromagnetic fields, superconducting quantum interference filters (SQIFs) show the typical behavior of nonlinear mixers. The SQIFs are manufactured from high-T c grain boundary Josephson junctions and operated in active microcooler. The dependence of dc voltage output V dc versus static external magnetic field B is nonperiodic and consists of a well pronounced unique dip at zero field, with marginal side modulations at higher fields. We have successfully exploited the parabolic shape of the voltage dip around B = 0 to mix quadratically two external time-harmonic rf-signals, at frequencies f 1 and f 2 below the Josephson frequency f J, and detect the corresponding mixing signal at |f 1− f 2|. When the mixing takes place on the SQIF current–voltage characteristics, the component at 2f 2−f 1 is present. The experiments suggest potential applications of a SQIF as a nonlinear mixing device, capable to operate at frequencies from dc to few gigahertz with a large dynamic range.

Regularity Theory and Superalgebraic Solvers for Wire Antenna Problems

We consider the problem of evaluating the current distribution $J(z)$ that is induced on a straight wire antenna by a time-harmonic incident electromagnetic field. The scope of this paper is twofold. One of its main contributions is a regularity proof for a straight wire occupying the interval $[-1,1]$. In particular, for a smooth time-harmonic incident field this theorem implies that $J(z) = I(z)/\sqrt{1-z^2}$, where $I(z)$ is an infinitely differentiable function—the previous state of the art in this regard placed I in the Sobolev space $W^{1,p}$, $p>1$. The second focus of this work is on numerics: we present three superalgebraically convergent algorithms for the solution of wire problems, two based on Hallén's integral equation and one based on the Pocklington integrodifferential equation. Both our proof and our algorithms are based on two main elements: (1) a new decomposition of the kernel of the form $G(z) = F_1(z) \ln\! |z| + F_2(z)$, where $F_1(z)$ and $F_2(z)$ are analytic functions on the real line; and (2) removal of the end-point square root singularities by means of a coordinate transformation. The Hallén- and Pocklington-based algorithms we propose converge superalgebraically: faster than $\mathcal{O}(N^{-m})$ and $\mathcal{O}(M^{-m})$ for any positive integer m, where N and M are the numbers of unknowns and the number of integration points required for construction of the discretized operator, respectively. In previous studies, at most the leading-order contribution to the logarithmic singular term was extracted from the kernel and treated analytically, the higher-order singular derivatives were left untreated, and the resulting integration methods for the kernel exhibit $\mathcal{O}(M^{-3})$ convergence at best. A rather comprehensive set of tests we consider shows that, in many cases, to achieve a given accuracy, the numbers N of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available; the required number M of integration points, in turn, can be several orders of magnitude smaller than those required in previous methods. In particular, four-digit solutions were found in computational times of the order of four seconds and, in most cases, of the order of a fraction of a second on a contemporary personal computer; much higher accuracies result in very small additional computing times.

An integral equation approach and the interior transmission problem for Maxwell's equations

In the first part of this paper we recall the direct scattering problem for time harmonic electromagnetic fields where arbitrary incident fields are scattered by a medium described by a space dependent permittivity, permeability, and conductivity. We present an integral equation approach and recall its basic features. In the second part we investigate the corresponding interior transmission eigenvalue problem and prove that the spectrum is discrete. Finally, we study the inhomogeneous interior transmission problem and show that it is uniquely solvable provided $k^2$ is not an interior eigenvalue.

Two-Tone Response in Superconducting Quantum Interference Filters

We successfully exploit the parabolic shape of the dc voltage output dip around B = 0 of a superconducting quantum interference filter (SQIF) to mix weak external r f signals. The two-tone response of weak time harmonic electromagnetic fields has been detected on the spectral voltage output of the SQIF at frequency f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> - f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> , for various frequencies f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> ranging from few MHz up to 20 GHz. The two-tone response is a characteristic function of static magnetic field B and of bias current I <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> , related to the second derivative of the dc voltage output.

Quadratic mixing of radio frequency signals using superconducting quantum interference filters

The authors demonstrate quadratic mixing of weak time harmonic electromagnetic fields applied to superconducting quantum interference filters (SQIFs), manufactured from high-Tc grain boundary Josephson junctions and operated in active microcooler. The authors use the parabolic shape of the dip in the dc voltage output around B=0 to mix quadratically two external rf signals, at frequencies f1 and f2 well below the Josephson frequency fJ, and detect the corresponding mixing signal at ∣f1−f2∣. Quadratic mixing also takes place when the SQIF is operated without magnetic shield. The experimental results are well described by a simple analytical model based on the adiabatic approximation.

Application of Diagonally Perturbed Incomplete Factorization Preconditioned Conjugate Gradient Algorithms for Edge Finite-Element Analysis of Helmholtz Equations

The diagonally perturbed incomplete factorization preconditioning scheme is applied to the conjugate gradient (CG) method for solving a large system of linear equations resulting from the use of edge-based finite-element method (FEM). This scheme contains more global information about the coefficient matrix when compared with banded-matrix schemes. The efficient implementation of this preconditioned CG (PCG) algorithm is described in detail for complex coefficient matrix equation. On several electromagnetic problems the PCG approach converges in CPU time, which is 8.6-19.5 times shorter with respect to the CG approach. By comparison with other preconditioned techniques, the results demonstrate that incomplete factorization preconditioning strategy is especially effective for CG iterative method when edge-FEM is applied to solve large-scale time-harmonic electromagnetic field problems

A FEM Domain Decomposition Method for Photonic and Electromagnetic Band Gap Structures

This paper introduces an efficient domain decomposition algorithm for the solution of time-harmonic electromagnetic fields arising in three dimensional, finite-size photonic band gap and electromagnetic band gap structures. The method is based on the finite element approximation and a nonoverlapping domain decomposition method. A set of "cement" unknowns on the inter-domain interfaces has been explicitly introduced to enforce the appropriate field continuities. The introduction of these extra unknowns allows for nonconforming/nonmatching triangulations across domain, eliminating the need for periodic mesh. To ensure and improve the convergence of the outer iteration loop, Robin transmission condition is used to communicate information across interfaces. The resulting system of equations is solved with a fast algorithm that loosely resembles the well known finite element tearing and interconnecting algorithm. In this algorithm, the method solves, in the preprocessing step, for the Robin primal subdomain problems multiple times, by exciting one dual unknown at a time. This step generates an iteration matrix that is then used to update the dual unknowns in the outer-loop iteration. The present method becomes extremely efficient for problems with geometric repetitions, such as, photonic and electromagnetic band gap structures.