A new numerical method for rough-surface scattering calculations

A new approach to solving the magnetic field integral equation (MFIE) for the current induced on a infinite perfectly conducting rough surface is presented. By splitting the propagator matrix into contributions from the left and from the right of the point of observation, a second kind integral equation can be formed with a new Born term and a new kernel. Following discretization of this new integral equation, the unknown currents can be determined more rapidly and with significantly less storage requirements than conventional LU decomposition; where the time saving factor is roughly N/3 where N is the number of current samples on the surface and the usual storage requirements associated with matrix inversion are eliminated. While the new Born term is usually adequate for scattered field calculations, the new discretized integral equation can be iterated to any desired accuracy with no apparent convergence problems. Results are presented for one-dimensional rough surfaces with rms heights exceeding one wavelength and rms slopes exceeding 40° which illustrate the robustness of the new Born term     

Application of the integral equation method of smoothing to random surface scattering

The integral equation method of smoothing (IEMS) is applied to the magnetic field integral equation (MFIE) weighted by the exponentialexp (jk_{1}zeta)wherezetais the stochastic surface height. An integral equation in coordinate space for the average of the product of the surface current and the exponential factor is developed. The exact closed-form solution of this integral equation is obtained based on the specularity of the average scattered field. The complex amplitude of the average scattered field is thus determined by an algebraic equation which clearly shows the effects of multiple scattering on the surface. In addition, it is shown how the incoherent scattered power can be obtained using this method. Comparisons with the Kirchhoff approximation and the dishonest approach are presented, and the first-order smoothing result is shown to be superior to both.     

Simplifications in the stochastic Fourier transform approach to random surface scattering

Further developments in the application of the stochastic Fourier transform approach (SFTA) to random surface scattering are presented. It is first shown that the infinite dimensional integral equation for the stochastic Fourier transform of the surface current can be reduced to the three dimensions associated with the random surface height and slopes. A three-dimensional integral equation of the second kind is developed for the average scattered field in stochastic Fourier transform space using conditional probability density functions. Various techniques for determining the transformed current (and, subsequently, the incoherent scattered power) from the average scattered field in stochastic Fourier transform space are developed and studied from the point of view of computational suitability. The case of vanishingly small surface correlation length is reexamined and the SFTA is found to provide erroneous results for the average scattered field due to the basic failure of the magnetic field integral equation (MFIE) in this limit.     

Hierarchal vector boundary elements and p-adaption for 3-Delectromagnetic scattering

This paper describes the development of new vector boundary elements for solving electromagnetic (EM) scattering problems. The new elements are suitable for the magnetic field integral equation (MFIE), electrical field integral equation (EFIE), or the combination field integral equation (CFIE). The basis functions are assigned to the edges of an element, rather than to its nodes. The new element guarantees the continuity of the normal component of the surface current across element edges. Furthermore, the basis functions are hierarchical from linear to higher order, which enables one to use the new elements in a p-adaption scheme     

On the testing of the magnetic field integral equation with RWGbasis functions in method of moments

For electromagnetic analysis using method of moments (MoM), three-dimensional (3-D) arbitrary conducting surfaces are often discretized in Rao, Wilton and Glisson basis functions. The MoM Galerkin discretization of the magnetic field integral equation (MFIE) includes a factor Ω₀ equal to the solid angle external to the surface at the testing points, which is 2π everywhere on the surface of the object, except at the edges or tips that constitute a set of zero measure. However, the standard formulation of the MFIE with Ω₀=2π leads to inaccurate results for electrically small sharp-edged objects. This paper presents a correction to the Ω₀ factor that, using Galerkin testing in the MFIE, gives accuracy comparable to the electric field integral equation (EFIE), which behaves very well for small sharp-edged objects and can be taken as a reference     

Contamination of the Accuracy of the Combined-Field Integral Equation With the Discretization Error of the Magnetic-Field Integral Equation

We investigate the accuracy of the combined-field integral equation (CFIE) discretized with the Rao-Wilton-Glisson (RWG) basis functions for the solution of scattering and radiation problems involving three-dimensional conducting objects. Such a low-order discretization with the RWG functions renders the two components of CFIE, i.e., the electric-field integral equation (EFIE) and the magnetic-field integral equation (MFIE), incompatible, mainly because of the excessive discretization error of MFIE. Solutions obtained with CFIE are contaminated with the MFIE inaccuracy, and CFIE is also incompatible with EFIE and MFIE. We show that, in an iterative solution, the minimization of the residual error for CFIE involves a breakpoint, where a further reduction of the residual error does not improve the solution in terms of compatibility with EFIE, which provides a more accurate reference solution. This breakpoint corresponds to the last useful iteration, where the accuracy of CFIE is saturated and a further reduction of the residual error is practically unnecessary.     

On the discretization of the integral equation describingscattering by rough conducting surfaces

Numerical simulations of scattering from one-dimensional (1-D) randomly rough surfaces with Pierson-Moskowitz (P-M) spectra show that if the kernel (or propagator) matrix with zeros on its diagonal is used in the discretized magnetic field integral equation (MFIE), the results exhibit an excessive sensitivity to the current sampling interval, especially for backscattering at low-grazing angles (LGAs). Though the numerical results reported in this paper were obtained using the method of ordered multiple interactions (MOMI), a similar sampling interval sensitivity has been observed when a standard method of moments (MoM) technique is used to solve the MFIE. A subsequent analysis shows that the root of the problem lies in the correct discretization of the MFIE kernel. We found that the inclusion of terms proportional to the surface curvature (regarded by some authors as an additional correction) in the diagonal of the kernel matrix virtually eliminates this sampling sensitivity effect. By reviewing the discretization procedure for MFIE we show that these curvature terms indeed must be included in the diagonal in order for the propagator matrix to be represented properly. The recommended current sampling interval for scattering calculations with P-M surfaces is also given     

An iterative current-based hybrid method for complex structures

This paper presents a general unified hybrid method for radiation and scattering problems such as antennas mounted on a large platform. The method uses a coupled electric-field integral equation (EFIE) and magnetic-field integral equation (MFIE) formulation, referred to as the hybrid EFIE-MFIE (HEM), in which the EFIE and MFIE are applied to geometrically distinct regions of an object. The HEM is capable of modeling arbitrary three-dimensional (3-D) metallic structures, including wires and both open and closed surfaces. We show that current-based hybrid techniques that utilize physical optics (PO) are an approximation of the HEM formulation. A numerical solution procedure is given that combines the moment method (EFIE) with an iterative Neumann series technique (MFIE). This permits one to effectively utilize the PO approximation when appropriate, and provides a general and systematic mechanism to correct the errors introduced by PO. Consequently, the HEM overcomes the inherent limitations of hybrid techniques which rely upon ansatz-based improvements of PO. The method is applied to the problem of radiation from objects that can be modeled using wires and metallic surfaces as fundamental elements. A representative example is given to demonstrate that the method can handle the difficult problem of a parasitic monopole located in the deep shadow region     

Linear-Linear Basis Functions for MLFMA Solutions of Magnetic-Field and Combined-Field Integral Equations

We present the linear-linear (LL) basis functions to improve the accuracy of the magnetic-field integral equation (MFIE) and the combined-field integral equation (CFIE) for three-dimensional electromagnetic scattering problems involving closed conductors. We consider the solutions of relatively large scattering problems by employing the multilevel fast multipole algorithm. Accuracy problems of MFIE and CFIE arising from their implementations with the conventional Rao-Wilton-Glisson (RWG) basis functions can be mitigated by using the LL functions for discretization. This is achieved without increasing the computational requirements and with only minor modifications in the existing codes based on the RWG functions     

The corrected induced surface current for arbitrary conductingobjects at resonance frequencies

It is well known that using the method of moments in conjunction with either the electric or magnetic field surface integral equations (EFIE or MFIE) produces inaccurate surface currents on conducting bodies at resonance frequencies. A new technique (based on singular value decomposition) is developed to correct the computed current by adding a correction factor term. This term is seen to be the resonant mode current, obtained by employing the power method in the moment method matrix, multiplied by an unknown complex factor. Applying the condition of vanishing field inside the conducting object results in obtaining the unknown complex factor. Therefore, this technique is hereafter referred to as correction factor technique (CFT). When the computed surface current on a conducting sphere, proposed technique, is compared with the exact one, the numerical results show excellent agreement     

A note on the use of the Neumann expansion in calculating thescatter from rough surfaces

The Neumann expansion has been used to compute the solutions of the magnetic-field integral equation (MFIE) for two-dimensional, perfectly conducting, Gaussian, rough surfaces. For surfaces whose roughness is of a similar order to the incident wavelength, it is shown that the expansion may diverge rapidly. The rate of convergence is compared with the conjugate-gradient (CG) method, whose convergence is sure. When it converges, the Neumann expansion convergence is more rapid. It is concluded that the Neumann expansion is not suitable without qualification as a numerical solution to the rough surface MFIE. Moreover, the failure of the Neumann expansion of the solution of the discrete representation of the MFIE provides strong evidence that the use of the Neumann expansion as a formal solution to the MFIE is open to doubt     

Analysis of electromagnetic scattering from conducting bodies ofrevolution using orthogonal wavelet expansions

The wavelet expansion method has been extended to study the electromagnetic scattering from conducting bodies of revolution. The magnetic field integral equation (MFIE) is solved by this approach. By expanding the induced surface currents in terms of Fourier series of uncoupled azimuthal cylindrical modes, a simplified MFIE is attained for each unknown mode current that varies along the curved profile of the scatterer. By applying the boundary element method (BEM), the curved profile is mapped into the definition domain of the orthogonal wavelets on the interval. The unknown mode currents are then expressed using multiscale wavelet expansions. The simplified MFIE is converted into a sparse, multilevel matrix equation by the Galerkin method. Numerical examples are provided to illustrate the merits of this wavelet approach     

A hybrid symmetric FEM/MOM formulation applied to scattering byinhomogeneous bodies of revolution

A new symmetric formulation of the hybrid finite element method (HFEM) is described which combines elements of the electric field integral equation (EFIE) and the magnetic field integral equation (MFIE) for the exterior region along with the finite element solution for the interior region. The formulation is applied to scattering by inhomogeneous bodies of revolution. To avoid spurious modes in the interior region a combination of vector and nodal based finite elements are used. Integral equations in the exterior region are used to enforce the Sommerfeld radiation condition by matching both the tangential electric and magnetic fields between interior and exterior regions. Results from this symmetric formulation as well as formulations based solely on the EFIE or MFIE are compared to exact series solutions and integral equation solutions for a number of examples. The behaviors of the symmetric, EFIE, and MFIE solutions are examined at potential resonant frequencies of the interior and exterior regions, demonstrating the advantage of this symmetric formulation     

A hybrid-iterative method for scattering problems

A new technique, named a hybrid-iterative method (HIM), is presented to solve the magnetic field integral equation (MFIE) for the induced currents on an arbitrary, perfectly conducting scatter. The technique is an evolution from two previous techniques developed earlier. The first of the previous techniques used the moment method (MM) to compute correction currents to an optics-type current. The second of the previous techniques effected a significant improvement by eliminating the use of the moment method to obtain the correction currents, using iteration to obtain them. The technique described here incorporates the edge diffraction theory and the Fock theory into the Ansatz of the iterative scheme. This procedure speeds up the algorithm as well as extending the range of problems that can be solved by the iterative scheme. Furthermore, the present technique incorporates the correction currents into the total currents thereby simplifying the iterative scheme. For intermediate size and larger bodies, the central processing unit (CPU) time is significantly less than that of the moment method. Results are presented for a variety of curved and edged two-dimensional cylinders illuminated by a transverse electric plane wave.     

An MFIE-based tabulated interaction method for UHF terrainpropagation problems

Approximations are introduced into a magnetic field integral equation (MFIE) formulation of a two-dimensional (2-D) terrain scattering problem, which allow most of the integrals inherent in the MFIE to be performed analytically. The implementation of the method is discussed and an example is given comparing its performance against a reference solution and measured data. The new formulation applies to both TM^z and TE^z polarizations and is an improvement over the electric field integral equation (EFIE) formulation of the tabulated interaction method (TIM) in that far-field patterns can be calculated analytically leading to increased flexibility of the method     

Improved testing of the magnetic-Field integral equation

An improved implementation of the magnetic-field integral equation (MFIE) is presented in order to eliminate some of the restrictions on the testing integral due to the singularities. Galerkin solution of the MFIE by the method of moments employing piecewise linear Rao-Wilton-Glisson basis and testing functions on planar triangulations of arbitrary surfaces is considered. In addition to demonstrating the ability to sample the testing integrals on the singular edges, a key integral is rederived not only to obtain accurate results, but to manifest the implicit solid-angle dependence of the MFIE as well.     

Time-domain BIE analysis of large three-dimensional electromagneticscattering problems

A time-domain boundary integral equation (BIE) solution of the magnetic field integral equation (MFIE) for large electromagnetic scattering problems is presented. It employs isoparametric curvilinear quadratic elements to model fields, geometry, and time dependence, eliminating staircasing problems. The approach is implicit, which seems to provide both stability and permits arbitrary local mesh refinement to model geometrically difficult regions without the significant cost penalty explicit methods suffer. Error dependence on discretization is investigated; accurate results are obtained with as few as five nodes per wavelength. The performance both on large scatterers and on low-radar cross section (RCS) scatterers is demonstrated, including the six wavelength “NASA almond,” two spheres, a thirteen wavelength missile, and a “high-Q” cavity     

The generalized forward-backward method for analyzing thescattering from targets on ocean-like rough surfaces

In Holliday et al. (1995, 1996), the iterative forward-backward (FB) method has been proposed to solve the magnetic field integral equation (MFIE) for smooth one-dimensional (1-D) rough surfaces. This method has proved to be very efficient, converging in a very small number of iterations. Nevertheless, this solution becomes unstable when some obstacle, like a ship or a large breaking wave, is included in the original problem. In this paper, we propose a new method: the generalized forward-backward (GFB) method to solve such kinds of complex problems. The approach is formulated for the electric field integral equation (EFIE), which is solved using a hybrid combination of the conventional FB method and the method of moments (MoM), the latter of which is only applied over a small region around the obstacle. The GFB method is shown to provide accurate results while maintaining the efficiency and fast convergence of the conventional FB method. Some numerical results demonstrate the efficiency and accuracy of the new method even for low-grazing angle scattering problems     

The use of curl-conforming basis functions for the magnetic-field integral equation

Divergence-conforming Rao-Wilton-Glisson (RWG) functions are commonly used in integral-equation formulations to model the surface current distributions on planar triangulations. In this paper, a novel implementation of the magnetic-field integral equation (MFIE) employing the curl-conforming n~/spl times/RWG basis and testing functions is introduced for improved current modelling. Implementation details are outlined in the contexts of the method of moments, the fast multipole method, and the multilevel fast multipole algorithm. Based on the examples of electromagnetic modelling of conducting scatterers, it is demonstrated that significant improvement in the accuracy of the MFIE can be obtained by using the curl-conforming n~/spl times/RWG functions.     

High-order numerical solutions of the MFIE for the linear dipole

The magnetic field integral equation (MFIE) was applied to a dipole using three different discretization methods and high-order basis functions. For moderate-order, and higher, basis functions it was found that the different discretization methods produced essentially the same results. Continuity of current and its first derivative was observed at cell boundaries even though continuity of current was not explicitly enforced there. The MFIE provided lower condition numbers than the Hallen equation over the range of dipole radii examined. In close proximity to surface discontinuities, including hidden ones, residual errors could not be significantly reduced by increasing the order of the basis functions, implying the need for better modeling at discontinuities and calling into question the use of faceting to represent curved surfaces.