A fast algorithm for electromagnetic scattering by buried three dimensional (3-D) dielectric objects of large size is presented by using the conjugate gradient (CG) method and fast Fourier transform (FFT). In this algorithm, the Galerkin method is utilized to discretize the electric field integral equations, where rooftop functions are chosen as both basis and testing functions. Different from the 3-D objects in homogeneous space, the resulting matrix equation for the buried objects contains both cyclic convolution and correlation terms, either of which can be solved rapidly by the CG-FFT method. The near-scattered field on the observation plane in the upper space has been expressed by two-dimensional (2-D) discrete Fourier transforms (DFTs), which also can be rapidly computed. Because of the use of FFTs to handle the Toeplitz matrix, the Sommerfeld integrals' evaluation which is time consuming yet essential for the buried object problem, has been reduced to a minimum. The memory required in this algorithm is of order N (the number of unknowns), and the computational complexity is of order N/sub iter/N log N, in which N/sub iter/ is the iteration number, and N/sub iter//spl Lt/N is usually true for a large problem.

Fast algorithm for electromagnetic scattering by buried 3-D dielectric objects of large size

The solution of the integral equation for a small width rectangular groove is considered. It is shown that by retaining the dominant mode supported by the rectangular groove, the resulting quasi-static integral equations are comparable to those associated with the perfectly conducting narrow strip. They are, therefore, amenable to analytic solution yielding the exact field distribution or equivalent currents across the groove's aperture. The derived currents exhibit the same edge behavior as that associated with the currents of a perfectly conducting half-plane. The corresponding current behavior based on a (numerical) impedance simulation of the groove is quite different. However the resulting echowidths are comparable. Both transverse electric (TE) and transverse magnetic (TM) polarizations are treated. >

Scattering from narrow rectangular filled grooves

A number of electromagnetic field problems for planar structures can be formulated in terms of a hypersingular integral equation, in which a grad-div operator acts on a vector potential. The vector potential is a convolution of the free-space Green's function and some surface current density over the domain of interest. A weak form of this integral equation is obtained by testing it with subdomain basis functions defined over the plate domain only. As a next step, the vector potential is expanded in a sequence of subdomain basis functions and the grad-div operator is integrated analytically over the plate domain only. For the problem of electromagnetic scattering by a plate, the method shows excellent numerical performance. The numerical difficulties encountered in some previous conjugate gradient fast Fourier transform (CGFFT) methods have been eliminated. >

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A weak form of the conjugate gradient FFT method for plate problems

The authors present generalized surface impedance conditions that improve medium characterization. One of the most common conditions that is often used to simulate a dielectric coating is the standard impedance boundary condition, and the improved boundary condition discussed can be viewed as a generalization of this, distinguished by the presence of higher-order derivatives of the field. By virtue of the derivatives, the conditions are less local in character, and the additional degrees of freedom can be used to better simulate the material properties of a surface. Following a review of the standard impedance condition, the new conditions are introduced. Several forms are presented, each with coefficients which are functions of the geometry and material properties of the surface. The applications of these to a variety of scattering problems are then discussed. >

Generalized impedance boundary conditions in scattering

An efficient numerical solution of the scattering by planar perfectly conducting or resistive plates is presented. The electric field integral equation is discretized using roof–top subdo–main functions as testing and expansion basis and the resulting system is solved via the biconjugate gradient (BiCG) method in conjunction with the fast Fourier transform (FFT). Unlike other formulations employed in conjunction with the conjugate gradient FFT (CG–FFT) method, in this formulation the derivatives associated with the dyadic Green's function are transferred to the testing and expansion basis, thus reducing the singularity of the kernel. This leads to substantial improvements in the convergence of the solution as demonstrated by the included results.

A Biconjugate Gradient FFT Solution for Scattering by Planar Plates

An application of a third-order generalized boundary condition to scattering by a filled rectangular groove is presented Deficiencies of such higher-order boundary conditions are addressed, and a correction for the present case is proposed As part of the process of examining and correcting the accuracy of the proposed generalized boundary conditions, an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition >

TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions

A technique to improve the convergence rate of the conjugate gradient-fast Fourier transform (CG-FFT) method is presented. The procedure involves the incorporation of subdomain basis functions associated with the current representation of linear and planar radiating elements. It is shown that significant improvements are achieved in the convergence of the CG-FFT when using sinusoidal basis functions. Numerical results are presented for thin cylindrical dipoles, conducting strips, and material plates of various sizes. In all cases, an increase in the rate of convergence by a factor of two or better was observed. >

Improving the convergence rate of the conjugate gradient FFT method using subdomain basis functions

Higher order boundary conditions involve derivatives of the fields beyond the first and were recently shown to be more effective than traditional first order conditions in modeling dielectric coatings and layers. In this report an application of a third order generalized boundary condition to scattering by a filled rectangular groove is presented. Deficiencies of such higher order boundary conditions are addressed and a correction is proposed for the present case. As part of the process of examining and improving the accuracy of the proposed generalized boundary conditions, an exact solution is developed and a comparison is provided with a solution based on the standard impedance boundary condition.

Scattering by a two dimensional groove in a ground plane

A technique is presented to improve the convergence of the conjugate-gradient fast Fourier transforms (CGFFT) method by the incorporation of subdomain basis functions associated with linear and planar radiating elements. To this end, the current is expressed as a convolution of a pulse basis with a finite sequence of delta functions whose amplitudes are the sampled values of the current. Using this current representation and the convolution theorem, the CGFFT formulation is extended to general basis functions. It is shown that significant improvements are achieved in the convergence of the method using sinusoidal basis functions. The improvement in the convergence is attributed to a more accurate approximation of the antenna current distribution. >

K. Barkeshli

Papers

Scattering from narrow rectangular filled grooves

TE scattering by a two-dimensional groove in a ground plane using higher order boundary conditions

Improving the convergence rate of the conjugate gradient FFT method using subdomain basis functions

Scattering by a two dimensional groove in a ground plane

Application of the conjugate gradient FFT method to large radiating systems using subdomain basis functions