Kadappan Panayappan

Singularity-free approach for the evaluation of the matrix elements in the context of the method of moments based on the use of closed-form expressions for the fields radiated by the subdomain basis functions

The Method of Moments (MoM) is the most commonly used technique for solving the Electric Field Integral Equation (EFIE) formulated for problems involving electromagnetic scattering from metallic objects. This formulation entails the expansion of the unknown current on the surface of the scatterer in terms of known basis functions [1, 2]. A key issue in the MoM solution is the evaluation of the singular integrals, which can be performed both by numerical and semi-analytical techniques [3]. A commonly used approach is to apply the singularity-extraction techniques, which is based upon subtracting the singular terms from the kernel and integrating them analytically. In this work, we propose a new method for the above field calculation, which utilizes the analytical expression for the near fields radiated by an electrically small dipole. The proposed method is found to be well suited for mitigating the problem of singularities encountered in the process of field computation while constructing the MoM matrix.

A Singularity Free MoM-Type of Formulation Using the Dipole-Moment-Based Approach (Invited Paper)

In this work we present a new physics-based approach for formulating MoM problems based on the use of dipole moments (DMs) — as opposed to the conventional Greens functions. The proposed technique is valid over the entire frequency range without any need for special treatments and is also free of singularities associated with the Greens function. The DM approach can be used equally well to both PEC and Dielectric objects. We also introduce certain refinements to the DM method to improve its computational efficiency like the use of higher-order basis functions, combining the DM with the Characteristic Basis Function Method (CBFM), the use of closed-form expressions for the calculation of interaction matrix elements and employing Fast Matrix Generation (FMG) for electrically large problems. We also demonstrate ways to incorporate lumped loads, capture sharp resonances even at low frequencies, calculate the input impedance of small antennas, calculate fields from irregular geometries; from faceted surfaces; from geometries with slot and slit; and also demonstrate the capability to model microstrip line type of geometries with fine features.

A universal and numerically efficient method of moments formulation covering a wide frequency band

In this paper, a novel MoM-based procedure is developed for efficient treatment of scattering problems over a wide frequency band, including very low frequencies where many of the conventional algorithms often fail. The proposed approach, based on closed-form expressions for the evaluation of the fields radiated by rooftop basis functions, is demonstrated to be universally applicable to conducting and dielectric objects without having to address the singularity issues associated with the conventional MoM approach. Furthermore, an equivalent rooftop type of basis function is introduced for handling curved surfaces. Illustrative examples demostrating the reliability and the efficiency of the described procedures are included.

On the hybridization of RUFD algorithm with the DM approach for solving multiscale problems

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.

A novel hybrid FDTD technique for efficient solution of multiscale problems

Numerical simulation of electromagnetic models with multi-scale features is highly challenging owing to the fact that electrically large as well as small features are simultaneously present in the model, which requires the computational domain to be discretized such that the number of degrees of freedom (DoFs) is very large; this, in turn, levies a heavy burden on the computational resources.

An Efficient Dipole-Moment-Based Method of Moments (MoM) Formulation

In this chapter, we present a technique for efficient derivation of the Method of Moments (MoM) matrix elements arising in radiation and scattering problems. The proposed method is designed to overcome some of the limits of the conventional MoM formulation, and it aims to provide a robust as well as efficient MoM-based approach. The chapter will begin by introducing a formulation based on the use of Dipole Moment (DM)-type of basis functions, and goes on to present the closed-form expressions for the fields radiated by the employed basis functions.

A new impedance boundary condition for FDTD mesh truncation

Summary form only given. It is well known that suitable absorbing boundary conditions (ABCs) are needed to truncate the mesh in the FDTD computational domain, and that the effectiveness of these boundary conditions affects the accuracy and efficiency of the FDTD simulations. The simplest and computationally inexpensive ABC is the MUR boundary condition (G. MUR, IEEE Trans. on Electromagnetic Compatibility, vol. 23, 1981). However, the accuracy of this boundary condition is good only for the normal incidence case, and the Mur ABC often causes reflections for oblique angles of incidence above an acceptable level. One of the widely used truncation condition is the Convoluted Perfectly Matched Layer, or more commonly known as CPML (W. Yu, R. Mittra, T. Su, Y. Liu and X. Yang, Parallel Finite-Difference Time Domain Method, Boston: Artech House, 2006). The performance of the CPML is superior to that of the MUR boundary condition, since the former is better able to suppress the reflections from the boundary; however, the CPML is computationally expensive. In this paper, we present an alternative to CPML by introducing a novel approach to FDTD mesh truncation, which is based on the use of an impedance boundary condition (IBC) for updating the fields at the boundaries of the computational domain. In this approach, the tangential E-Fields at the end of the computational domain are calculated from the H-fields based on the impedance relationship: Etan = ηn × H where η is a suitably chosen impedance value. The H-fields at the boundaries of the computational domain are updated by using the conventional FDTD equations, but the E-fields are derived by using the IBC. The results presented below illustrate the accuracy of the proposed algorithm, which requires much less CPU time and memory than those needed by the CPML.

Formulating matrix equations in the context of MoM by using the Dipole Moment (DM) method instead of Green's functions

In this paper we introduce a universal Method of Moment (MoM)-based formulation, which overcomes some of the drawbacks of conventional frequency domain techniques. Formulating the problem in a way that bypasses the evaluation of the electric field as a summation of scalar and vector potential terms enables us to overcome the low-frequency breakdown when applied to the solution of the Surface Electric Field Integral Equation (S-EFIE). The direct evaluation of the electric fields reduces the impedance matrix fill time compared to the conventional MoM. The Equivalent Medium Approach (EMA) can be applied in conjunction with the proposed formulation for the solution of microstrip circuits embedded in stratified environments.

Low Frequency Modelling of Layered Media for Logging While DrillingApplications Using FDTD

In a typical Logging While Drilling (LWD) ap- plication, several coils operating in the frequency range of a few KHz to MHz are used as transmitters and receivers to appropriately characterize the earth formation. Electromagnetic modelling of such a low frequency system poses serious computational challenges. In the Method of Moment (MoM) formulation, contribution of vector potential to the total field becomes several orders of magnitude smaller than that of the scalar potential, thus making the resultant matrix highly ill-conditioned. Finite Difference Time Domain (FDTD) method, on the other hand, requires enormous number of time steps to capture the low frequency information. In this paper, we consider a layered-earth model and compute the electromagnetic field due to electric and magnetic dipoles embedded in the formation. To address the low frequency problem in FDTD, we consider the source and the receiver dipoles to be infinitesimally small and aligned with the computational grid, and we modify the update equations accordingly. This approach reduces the time convergence of FDTD by two-to-three orders of magnitude, and also reduces the memory requirements by the same factor. Numerical results for the fields reflected from the layered interfaces and the corresponding voltages induced in the receive coils are presented for multiple scenarios involving shale and sand zones.

A technique for solving multiscale problems by hybridizing frequency and time domain algorithms

Summary form only given. Despite the availability of modern supercomputers, direct solution of multiscale problems by means of conventional CEM methods-be it FEM, FDTD or MoM-is highly challenging. This is because modeling of structures with fine features, which might share the computational domain with other large objects, often requires dealing with a large number of degrees of freedom. Dealing with such multiscale problems often forces us to compromise the accuracy of the numerical discretization process when faced with the problem of capturing the small-scale features, in order to cope with the limited available resources in terms of CPU memory and time.Many fine-featured structures that are integral parts of sensors and other complex systems have dimensions that are often only a small fraction of the wavelength in the medium and require a very fine mesh to capture the nuances of their geometry. In the past, it has been demonstrated that the electromagnetic properties of small objects can be accurately and conveniently characterized by a Dipole Moment (DM) representation, which is valid both in the nearand far-field regions of the scatterer. Based on this, a hybrid FDTD method has been proposed, which models the fine features by using a DM located at the center of the FDTD cell. Since the DM is always located in close vicinity of the grid lines of the cell containing the DM, the 1/r3 term dominates within the cell, rendering the fields “time independent” and identical to those from a static charge. These fields are then passed on to the FDTD, which automatically performs the analytic continuation of the quasi-static solution into the region external to the cell containing the small object, as a consequence of the dynamic nature of Maxwells equations. However, when modeling straight wires passing through multiple FDTD cells, the above hybrid method fails since the quasi-static approximation of the DM is no longer valid over the entire wire. To address this problem, we propose to model the wire by using basis functions which can be conveniently expressed in closed forms, not only in the frequency domain, but in the time domain as well. The scheme is simpler and also computationally more efficient than the conventional Time Domain Integral Equation (TDIE) algorithms, especially when a large number of unknowns is involved. However, we have found that the proposed algorithm becomes unstable when we generalize the field expressions in order to model bent wire geometries, for which a time domain basis function which is convenient, causal and stable cannot be found. In this work we propose a novel DM-FDTD hybrid technique when dealing with an arbitrarily shaped wire, which may span several cells. The key is to use a modified DM approach, in which all the three terms of the fields generated by the DM, corresponding to both the near and far-field regions, are expressed in the time domain in convenient closed forms. Thus, the field contributions can be calculated at any arbitrary distancefrom the DM source and can be coupled with the FDTD. The proposed method is universal and can be used for scatterers traversing through multiple cells. Also, since the fields produced from an arbitrary scatterer can always be represented as a superposition of electric and/or magnetic DMs, the method can be employed to model an arbitrarily shaped thin scatterer without altering the FDTD updating scheme or the size of the FDTD discretization, which can be the nominal λ/20, even when dealing with objects with fine features that are small fractions of the wavelength.