Error-controllable and well-conditioned mom solutions in computational electromagnetics: ultimate surface integral-equation formulation [open problems in cem

A novel surface integral equation (IE) formulation for magneto-quasi-static analysis of current flow in two-dimensional conductors situated in lossy layered substrates is proposed. Traditional approach is based on the volume IE formulated with respect to the unknown current distribution in the conductors cross-section. In this work, we transform the volume IE to a surface IE via representation of the volumetric current density as a superposition of the waves emanating from the conductors boundaries. The resulting surface IE features a single unknown surface current density unlike the traditional surface IEs which require both electric and magnetic surface current densities. Also, the resultant single-source surface IE formulation involves only the electric field Greens functions instead of both electric and magnetic field Greens functions as required by the traditional surface IE formulations. The latter property makes the proposed IE formulation particularly suitable for analysis of current flow in conductors embedded in lossy layered substrates.

Surface-integral-equation (SIE) approaches are presented for the accurate solution of different problems in computational electromagnetics. First, an efficient message passing interface (MPI)/OpenMP parallel implementation of the multilevel fast multipole algorithm-fast Fourier transform (MLFMA-FFT) is presented for the solution of large-scale conducting bodies. Second, looking for the extension of these rigorous approaches to new demanded areas, the SIE method is successfully applied to the solution of left-handed metamaterials and plasmonic nanostructures.

The Origin of Spurious Solutions in Computational Electromagnetics

The surface integral equation method is one of the most popular numerical methods for the electromagnetic analysis of composite metallic and dielectric structures. In this paper, the authors seek the "optimal" surface integral equation formulations for closed metallic, homogeneous and piecewise homogeneous dielectric and composite objects by studying the properties of the integral equations. The goal is to develop formulations in which all unknown quantities are well tested in the Galerkin method and the discretized matrix equation has good spectral properties leading to rapidly converging iterative solutions. The results show that by deriving the equations so that they map the currents onto itself and are directly tested in the same space as the unknowns are expanded can significantly improve conditioning of the matrix equation.

Application of the pseudospectral method in numerical calculations is well established. The method is used in several disciplines, from geology to applied mathematics. The pseudospectral method also appears in computational electromagnetism. Its implementation already has been carried out for the finite element method and finite difference time domain method. In this study, we implement the pseudospectral method for the finite difference frequency domain (FDFD) method, and hence the resulting method is called the pseudospectral frequency domain (PSFD) method. The PSFD method is compared to the FDFD method in terms of the numerical phase velocity and anisotropy. The method is extended for the nonlinear process PSFD (NL-PSFD) of second-harmonic generation. It is shown how a plane wave source at oblique incidence can be implemented, and its performance for tilted quasi-phase-matched grating is discussed. Finally, a specific method is deduced from NL-PSFD for large-volume simulation, where only second-order nonlinearity is spatially structured, and a two-dimensional nonlinear photonic crystal is simulated.

The volume integral equation (VIE) and the surface integral equation (SIE) formulations of the resonant mode, frequency, and quality factor of arbitrarily shaped dielectric resonators (DR) in free space and in a semi-open multilayered medium are compared. Although both methods are very efficient tools for resonant field/frequency determination for microwave circuit design and analysis, we have found that SIE is not suitable for determining the quality factor of high quality DRs of interest in high performance microwave filters and oscillators. As the quality factor goes beyond several hundred, the quality factor obtained through SIE-MoM becomes erroneous. It may fluctuate randomly depending on the mesh. However, VIE-MoM has been found to be robust and reliable for DRs with very high quality factors.

The standard Yee FDTD algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of this classical scheme to 3D unstructured co-volume meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This circumvents the problem of accuracy losses, which are normally associated with the use of a staircased representation of curved material interfaces in the standard Yee scheme. The procedure has been successfully employed for modelling problems involving both isotropic and anisotropic lossy materials. Here, we consider the novel extension of this approach to allow for the challenging modelling of chiral materials, where the material parameters are frequency dependent. To adequately model the dispersive behaviour, the Z-transform is employed, using second order Padé approximations to maintain the accuracy of the basic scheme. To validate the implementation, the numerical results produced are compared with available analytical solutions. The stability of the chiral algorithm is also studied.

Maxwell's equations are valid from the atomic to intergalactic length scales. Hence, features at various spatial or temporal scales may all be significant in electromagnetic (EM) problems of interest and may affect the results of multiscale modeling and simulation in computational electromagnetics (CEM). Solving electromagnetic problems that involve multiscale structures efficiently is challenging and is of growing interest in CEM. In recent decades, various approaches have been proposed for addressing the multiscale issue of electrically large problems. These methods can be roughly categorized into three types: the “geometry‐based,” “algebra‐based,” and “physics‐based” approaches. In this chapter, we introduce these three representative approaches for modeling multiscale problems in CEM: (1) the discontinuous Galerkin integral equation method (DGIE), which is a general domain decomposition framework for dealing with structures with mesh nonconformity; (2) the nested dissection accelerated H‐LU‐based fast direct solver for sparse matrices, which is used to construct effective preconditioners; and (3) the resistive sheet boundary condition‐based hybrid finite element‐boundary integral‐multilevel fast multipole algorithm (FE–BI–MLFMA), which can simulate electromagnetic field interactions with microwave‐absorbing honeycomb structures efficiently. After we present the formulation for each approach, we present examples to demonstrate its accuracy, capability, and performance.

In addition to its technical merits as a challenging nonrigid motion and structural integrity analysis problem, quantitative estimation of cardiac regional functions and material characteristics has significant physiological and clinical value. We developed a stochastic finite-element framework for the simultaneous recovery of myocardial motion and material parameters from medical image sequences with an extended Kalman filter approach, and we have shown that this simultaneous estimation strategy achieves more accurate and robust results than separated motion and material estimation efforts. In this paper, we present a new computational strategy for the framework based upon the maximum a posteriori estimation principles, realized through the extended Kalman smoother, that produces a sequence of kinematics state and material parameter estimation of the entire myocardium, including the endocardial, epicardial, and midwall tissues. The system dynamics equations of the heart are constructed using a biomechanical model with stochastic parameters, and the tissue material and deformation parameters are jointly estimated from the periodic imaging data. Noise-corrupted synthetic image sequences with known kinematics and material parameters are used to assess the accuracy and robustness of the framework. Experiments with canine magnetic resonance tagging and phase-contrast image sequences have been conducted with very promising results, as validated through comparison to the histological staining of postmortem myocardium.

This article presents a unified description of numerical methods for solving electromagnetic field problems. Traditionally, these methods are considered to be independent and alternative ways of solving Maxwell’s equations or equations derived therefrom. However, they all are projective approximations of the unknown solution by known expansion or basis functions with unknown amplitude coefficients that are determined using the so- called method of weighted residuals (MWR). This common feature forms the proposed unifying framework that may serve both as a systematic introduction to computational electromagnetics and as a way to provide insight into the nature, strengths, and limitations of the various algorithms used by present and future field solvers. For convenience, the fundamental equations and concepts of electromagnetics are summarized in order to facilitate the demonstration of the unified approach. Our aim is to provide students, designers, and researchers with a framework for developing new numerical algorithms, to guide users in the selection and application of electromagnetic design tools, and to foster informed engineering judgment. This article also serves as a review and summary of earlier theoretical works reported in different places and at different times.

There exists a great need to develop numerical techniques for efficient solution of multiscale electromagnetic problems, regardless of the computational electromagnetics (CEM) method currently being used to tackle them, be it FEM, FDTD or MoM. Dealing with multiscale objects, which have features that are both small and large compared to the wavelength, is highly challenging and often forces us to compromise the accuracy (relaxing the numerical discretization process when attempting to capture the small-scale features) in order to cope with the limited available resources in terms of CPU memory and time. In this paper, we introduce a new scheme that combines the Recursive Update Frequency Domain (RUFD) method with the Dipole Moment (DM) method, to solve multiscale problems in a numerically efficient manner.

PurposeThis paper aims to review various techniques used in computational electromagnetism such as the treatment of open problems, helicoidal geometries and the design of arbitrarily shaped invisibility cloaks. This seemingly heterogeneous list is unified by the concept of geometrical transformation that leads to equivalent materials. The practical set‐up is conveniently effected via the finite element method.Design/methodology/approachThe change of coordinates is completely encapsulated in the material properties.FindingsThe most significant examples are the simple 2D treatment of helicoidal geometries and the design of arbitrarily shaped invisibility cloaks.Originality/valueThe paper provides a unifying point of view, bridging several techniques in electromagnetism.

Effect and its mechanism of spatial coherence of track irregularity on dynamic responses of railway vehicles

Semiconductors with higher carrier mobility and carrier density are required to fabricate a p-n junction diode for high-speed device operation and high-frequency signal processing. Here, we use a chemically doped semiconducting single-walled carbon nanotube (SWCNT) random network for a field effect transistor (FET) and demonstrate a rectifier operated at a wide range of frequencies by fabricating a p-n junction diode. The p-n diode was fabricated by using a pristine p-type SWCNT-FET where half was covered by SiO(2) and the other half was chemically doped by using benzyl viologen molecules, which was converted into an n-type channel. The half-wave rectifier of the random network SWCNT p-n junction diode clearly highlights the device operation under high input signal frequencies up to 10 MHz with very low output distortion, which a commercial silicon p-n junction diode cannot access. These results indicate that the random network SWCNT p-n junction diodes can be used as building blocks of complex circuits in a range of applications in microelectronics, optoelectronics, sensors, and other systems.

Enforcement of current continuity at material junctions in surface integral equation (SIE) formulations for composite objects has generally relied on somewhat intrusive constraint procedures that complicate system assembly and preconditioning, especially in the context of low rank solvers. The method proposed alleviates these problems by enforcing constraints in a manner equivalent to earlier approaches, but with a less invasive implementation. In addition to solving these challenges, the method is sufficiently general to handle other types of constraints, e.g., unknowns in symmetry planes.