Efficient method of moments formulation for large PEC scattering problems using asymptotic phasefront extraction (APE)

Efficient analysis of electromagnetic-wave responses from electrically large complex objects is a crucial but difficult topic in many applications of electromagnetic engineering. Among various numerical methods, the Method of Moments (MoM) solves for surface- or volume-distributed equivalent currents, which are expanded in terms of basis functions. In this paper, the traveling-wave phase variation has been incorporated into the basis functions to enhance their capability of describing the equivalent surface currents induced over surfaces of perfect electric conductors (PECs). A rigorous derivation is first given as physical insight, to show that the induced surface current is composed of a traveling-wave term and several standing-wave terms. A phase-extracted basis function (PEBF) is then proposed to describe the traveling current wave, and is applied in solving the electromagnetic scattering from three-dimensional (3D) PEC objects with smooth surfaces. By hybridizing the phase-extracted basis function with higher-order hierarchical basis functions, a moving standing-wave (MSW) basis function is further introduced to describe both the traveling-wave and the standing-wave terms in the induced surface current. It is shown from several numerical examples that the moving standing-wave basis function has excellent performance for solving electromagnetic scattering from three-dimensional PEC objects with both smooth and non-smooth surfaces, as well as for three-dimensional PEC cavities. After that, the unique properties of the phase-extracted basis function are utilized in the “sparsification” of the MoM matrix, and in the fast calculation of wideband responses. With the aid of the phase-extracted basis function and the moving standing-wave basis functions, the total memory requirements and the computational time can be significantly reduced, while numerical solutions can be obtained with good accuracy. The advantages of the phase-extracted basis functions and the moving standing-wave basis functions in predicting electromagnetic scattering from electrically large complex objects are summarized before the conclusions are drawn.

Printed circuits in bounded media encompass a wide range of practical structures such as discontinuities in waveguides, planar circuits embedded in shielded multilayered media or even two-dimensional printed periodic structures. The Electromagnetic (EM) modeling of printed circuits in layered bounded media is performed via an Integral Equation (IE) technique. Green's functions (GFs) are specially defined to satisfy both the Boundary Conditions (BCs) imposed by the layered media and by the transverse boundary enclosing the entire structure. Finally, a system of IEs on the equivalent sources can be solved numerically by means of the Method of Moments (MoM). Each of the problems enumerated above has already been solved by other authors using IE-MoM techniques. Nevertheless, our formulation introduces a unified approach applicable to all the aforementioned problems. Due to the symmetry presented by a bounded layered media, the GF problem can be reduced into a two-dimensional transverse boundary problem and a one-dimensional transmission line problem in the normal direction. Both problems can be treated independently. This thesis proposes and fully develops an efficient technique that encompasses different laterally bounded multilayered problems with a seamless transition between them. The method is based on a modal representation of the transverse boundary problem and on the expansion of the equivalent surface currents by zero-curl & constant-charge Basis Functions (BFs). It offers a unified and versatile approach that, on one hand eliminates redundancy in the formulation and on the other hand simplifies each particular problem to the evaluation of constant coefficients or basic line integrals. Analytical solutions can be found for the combination of linear subsectional basis functions in rectangular and circular Perfect Electric Conductor (PEC) boundaries as well as for periodic lattices. This thesis then solves the problem of transmission line model in the longitudinal direction by proposing an efficient algorithm that guarantees numerical stability under a variety of known critical conditions where other already known formulations fail. In addition, it introduces alternate equivalent expressions of this formulation that allow new interpretations of the problem. Due to its practical interest, the method is applied for the EM modeling of multilayered boxed printed circuits. This motivated the implementation of a dedicated software tool for the efficient analysis of these topologies including losses. Extensive numerical experiments have been carried out to assess the validity of the aforementioned theory and some properties of test-structures (losses, mesh, etc).

A simple volume and surface formulation is presented to analyze the problem of scattering of TE waves by multiple perfectly conducting and dielectric cylinders of arbitrary cross section. The Method of Moments and the principle of equivalence are used to replace the cylinders by equivalent surface and volume currents radiating in an unbounded medium. The surface currents are obtained by enforcing the boundary conditions on the surface of the conducting cylinders, and the volume currents are obtained by applying the equivalence principle inside the volume of the material cylinders. This procedure leads to integral equations for the currents in each of the cylinders considered. The Method of Moments is used to solve the integral equations. Triangular functions are used to expand the surface currents on the conducting cylinders and a set of volume solenoidal basis functions is used to expand the volume currents in the dielectric cylinders. Some of the deficiencies of the formulation are discussed. Results ar...

A nonconformal scheme for scattering analysis from PEC objects

Plane wave scattering by a cylindrically conformal periodic finite array (CCPFA) with perfectly conducting (PEC) and resistive patches, including graphene patches are studied. CCPFA represents a dielectric rod (DR) with patches placed in azimuth direction. The problem solution was obtained by the Method of Moments (MoM) applied in the spectral domain using Piece Wise Sinusoidal (PWS) basis functions. The structure geometries in the case when the PEC patches are nearly touching are carefully analyzed. It was found in the discussed 3D case that for a half wavelength resonance the natural frequency tends to zero when the slot is decreasing (gap-coupled case). Also it was shown that it is in a very good agreement with 2D analytical solution until the slot angular size is less than 40°. For CCPFA with resistive, including graphene patches the patch resistivity influence on the resonant frequency is discussed. The current distribution, far fields and the resonant frequency behavior versus frequency are presented.

Electromagnetic (EM) scattering from perfect electric conductor (PEC) object embedded between two rough interfaces is investigated utilizing method of moments (MoM) for both TE and TM polarizations. Theoretical formulations of EM scattering model considered in this paper is presented and the total fields and their derivatives are solved numerically by employing MoM. The rough surfaces are modeled as realizations of a Gaussian random process with the Gaussian spectrum, while the tapered incident wave is chosen to reduce the truncation error. The influences of characteristic parameters of the rough surfaces, the relative permittivity of the medium, as well as the average height between the two rough surfaces, on the bistatic scattering coefficient (BSC) are discussed.

The power-loss method, along with a surface integral formulation, has been used to compute the attenuation constant in microstrip and coplanar structures. This method can be used for the analysis of both open and closed structures. Using the surface equivalence principle, the waveguide walls are replaced by equivalent electric surface currents and dielectric surfaces are replaced by equivalent electric and magnetic surface currents. Enforcing the appropriate boundary condition, and E-field integral equation (EFIE) is developed for these currents. Method of moments with pulse expansion and point matching testing procedure is used to transform the integral equation into a matrix one. The relationship between the propagation constant and frequency is found from the minimum eigenvalue of the moment matrix. The eigenvector pertaining to the minimum eigenvalue gives the unknown electric and magnetic surface currents.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The diffracted field due to an E-polarized plane wave incident on two cylindrical wedges forming a slit-type aperture is formulated using an integral equation for the surface current. The use of the ordinary method of moments to solve for the unknown surface currents does not yield a convergent solution, owing to the semi-infinite dimensions of the wedges and the fact that the total current does not vanish on the illuminated surface for this type of excitation. Therefore, the surface current is divided into two components, namely, a known noninteraction term and an unknown term due to the infinite number of interactions between the wedges. The noninteraction term is considered to be the physical optics current for those geometries that have wedges with nonsharp edges, whereas the exact surface current for a single wedge is used for double wedges with sharp edges. The interaction term may be evaluated using the ordinary method of moments, because it decays with increasing distance from the edge along the surface of either wedge. It is found that this technique leads to an efficient solution and makes it possible to use the method of moments for the multiple scattering by bodies of semi-infinite or large dimensions. Numerical results for the diffracted fields and transmission coefficients for a variety of related structures, such as the thin slit, the thick slit, the double truncated wedge, and the double capped wedge, are presented and compared with available solutions.

In realistic landmine fields, the anti-personnel plastic mine is often buried nearby a clutter-object under the ground. The presence of a second object buried near the mine under a two-dimensional (2-D) rough ground can easily obscure the target and/or cause a false alarm. The separation distance between the AP mine and clutter-object plays a significant role on the probability of true or false alarm in this situation. A rigorous electromagnetic model has been developed to analyze the scattering mechanism between the target and the clutter-object, between the target and the rough ground, between clutter-object and the rough ground and the multiple scattering between different spots on the rough ground itself. The new rigorous model is based on the classical electromagnetic equivalence theorem leading to producing six new integral equations. Using the Method of Moment (MoM), the new integral equations are transformed into a linear system of equations to be solved for the unknown electric and magnetic currents on the surface of three scatterers; 1) rough ground, 2)target and 3)clutter-object. The MoM impedance matrix completely represents every interaction between these three scatterers. The superior Steepest Descent Fast Multipole Method (SDFMM) is used to tremendously speed up the computations of the unknown MoM surface currents.

A general numerical solver for analyzing microstrip geometries of rectangular shape is presented in this paper. The analytical foundation of this solver is based on an integral equation approach which we formulate in the spatial domain. The unknown surface current on the microstrip is solved by the method of moments using 2D rectangular pulses as the expansion functions. Transmission‐line modeling is then used to parameterize a given microstrip junction or discontinuity. Aided by a graphics interface, this solver can analyze complex structures without incurring additional analytical complexity. We illustrate the accuracy and versatility of our solver by applying it to several different microstrip discontinuities ranging from a single‐stub to an interdigitated capacitor.

Electromagnetic (EM) problems with conducting and penetrable media are formulated by volume-surface integral equations (VSIEs) which allow inhomogeneous and anisotropic materials. Traditionally, the VSIEs are solved by the method of moments (MoM) with Rao-Wilton-Glisson (RWG) and Schaubert-Wilton-Glisson (SWG) basis functions. We propose a point-matching scheme to discretize the VSIEs. The point-matching scheme does not rely on any basis and testing functions and allows the use of nonconforming meshes. Also, the integral kernels are free of material parameters by selecting current densities as unknowns. To accelerate the solution for electrically large problems, we incorporate the scheme with the multilevel fast multipole algorithm (MLFMA) and a typical numerical example is presented to validate its effectiveness.

The extension of Rao, Wilton and Glisson basis functions on flat-faceted triangular elements to different element shapes, higher-order geometry and higher-order function support is outlined. A curvilinear coordinate formulation is used to obtain a family of finite elements for more accurate method of moments (MoM) computations. Results for the first two orders of geometry and function support in 1, 2 and 3 dimensions are presented. The basis functions are hierarchical, in that mixed orders of geometry and of function support can be used together in a single calculation to allow efficient local refinement. Practical issues of element assembly, treatment of singular and non-singular MoM integrals and of loop basis functions are addressed.

The method of moments (MOM) in conjugation with the best uniform rational approximation is applied to predict the broad-band radar cross-section of arbitrarily shaped three-dimensional (3D) bodies. The surface integral equations are solved using MOM to obtain the equivalent surface currents at the frequency points corresponding to the Chebyshev nodes, and then the surface current within the given frequency band is represented by the Chebyshev series. To improve the accuracy, the Chebyshev series is matched via the Maehly approximation to a rational function, which can be as good as the best uniform rational approximation. Using the rational function, the surface current is obtained at any frequency point within the frequency band. Numerical results for 3D arbitrarily shaped perfectly electric conducting objects and homogenous dielectric bodies are considered. Compared with the asymptotic waveform evaluation technique and the model-based parameter estimation, the proposed technique is accurate in much broader frequency band with lower memory required.

We present scattering computations performed with a newly developed stable hybridization of the finite element method (FEM) and the finite-difference time-domain (FDTD) scheme, which is based on Nitsche's method. This hybrid has not been tested on scattering problems previously, and here we compute the radar cross-section (RCS) for three different targets: (i) the perfect electric conducting (PEC) sphere, (ii) the NASA almond, and (iii) a generic aircraft called RUND. In order to assess the discretization errors associated with the hybrid, we provide comparisons with established results and techniques: (i) the Mie-series for the PEC sphere, (ii) the method of moments (MoM) implemented in the commercial code FEKO, and (iii) a stable FEM–FDTD hybrid that exploits pyramids and a curl-conforming representation of the electric field. The bistatic RCS for the PEC sphere shows second order convergence towards the analytical solution and a relative error of 2% is achieved for about 20 points per wavelength. The NASA almond has a low monostatic RCS in its horizontal plane, which makes it a good benchmark problem for scattering computations. It features a sharp tip and, as a consequence, the order of convergence for the RCS is lowered. Given a careful convergence study for the NASA almond, we achieve a highly accurate monostatic RCS by means of extrapolation and this result is state-of-the-art in the open literature.

We address in detail the numerical simulations of electromagnetic (EM) wave scattering from a mockup partly coated fighter jet in this presentation. The majority part of the fighter jet surface is considered to be perfect electric conductor (PEC). The horizontal stabilizer, vertical stabilizer and the wings are coated with high-density materials, whereas a hollow dielectric radome is located at the front of the aircraft. Due to the nature of unbounded problem domain, and the desire to avoid a three dimensional discretization, we attempt to compute the numerical approximation using the boundary integral equations (BIEs), or the method of moments (MoM). However, it turns out not to be an easy quest. In the presentation, we shall elucidate difficulties that we have encountered and innovative techniques that we have devised during our journey to solve such a challenging mutli-scale EM application.