Shu-Qing Li

A sparse-matrix/canonical grid method for analyzing densely packed interconnects

In this paper, a fast numerical method called the sparse-matrix/canonical-grid (SM/CG) method is employed to analyze densely packed microstrip interconnects that involve a large number of unknowns. The mixed-potential integral equation is solved by using the method of moments in the spatial domain. The closed-form expressions of the spatial Greens functions of microstrip structures are obtained from the combination of the fast Hankel transform and the matrix pencil method. The Rao-Wilton-Glisson triangular basis functions are used to convert the integral equation into a matrix equation. The matrix equation is then solved by using the SM/CG method, in which the far-interaction portion of the matrix-vector multiplication in the iterative solution is performed by the fast Fourier transforms (FFTs). This is achieved by the Taylor series expansions of the spatial Greens functions about the uniformly spaced canonical grid points overlaying the triangular discretization. Numerical examples are presented to illustrate the accuracy and efficiency of the proposed method. The SM/CG method has computational complexity of O(NlogN). Furthermore, being FFT-based facilitates the implementation for parallel computation.

An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method

An efficient algorithm for wave scattering from two-dimensional lossy rough surfaces is proposed. It entails the use of a single magnetic field integral equation (SMFIE) in conjunction with a multilevel sparse-matrix canonical-grid (MSMCG) method. The Rao-Wilton-Glisson (RWG) triangular discretization is adopted to better model the rough surface than the pulse basis functions used in the well-established SMCG method. Using the SMFIE formulation, only one unknown per interior edge of the triangular mesh approximating the rough surface is required, and the iterative solution to the moment equation converges more rapidly than that of the conventional coupled equations for dielectric rough surfaces. The MSMCG method extends the applicability of the SMCG method to rougher surfaces. Parallel implementation of the proposed method enables us to model dielectric surfaces up to a few thousand square wavelengths. Simulation results are presented as bistatic scattering coefficients for Gaussian randomly rough surfaces.

Parallel implementation of the sparse-matrix/canonical grid method for the analysis of two-dimensional random rough surfaces (three-dimensional scattering problem) on a Beowulf system

Wave scattering from two-dimensional (2-D) random rough surfaces [three-dimensional (3-D) scattering problem] has been previously analyzed using the sparse-matrix/canonical grid (SM/CG) method. The computational complexity and memory requirement of the SM/CG method are O(N log N) per iteration and O(N), respectively, where N is the number of surface unknowns. Furthermore, the SM/CG method is FFT based, which facilitates the implementation on parallel processors. In this paper, we present a cost-effective solution by implementing the SM/CG method on a Beowulf system consisting of PCs (processors) connected by a 100 Base TX Ethernet switch. The workloads of computing the sparse-matrix-vector multiplication corresponding to the near interactions and the fast Fourier transform (FFT) operations corresponding to the far interactions in the SM/CG method can be easily distributed among all the processors. Both perfectly conducting and lossy dielectric surfaces of Gaussian spectrum and ocean spectrum are analyzed thereafter. When possible, speedup factors against a single processor are given. It is shown that the SM/CG method for a single realization of rough surface scattering can be efficiently adapted for parallel implementation. The largest number of surface unknowns solved in this paper is over 1.5 million. On the other hand, a problem of 131072 surface unknowns for a PEC random rough surface of 1024 square wavelengths only requires a CPU time of less than 20 min. We demonstrate that analysis of a large-scale 2-D random rough surface feasible for a single realization and for one incident angle is possible using the low-cost Beowulf system.

Microwave emission of rough ocean surfaces with full spatial spectrum based on the multilevel expansion method

Microwave emission of ocean surfaces with full spatial spectrum is studied in this paper. For ocean surfaces with full spectrum, the rms height of roughness can be many wavelengths, and the surface size must be chosen to be larger than the longest scale wave in the spectrum. Due to computer resources, it is not straightforward to conduct numerical simulations of emission from rough surfaces with large rms height and size since a large number of unknowns will be involved. In this paper, the multilevel expansion of the sparse matrix canonical grid (SMCG) method, which is available for surfaces with large rms heights, is used to study the emission of one-dimensional (1-D) ocean surfaces. The computational complexity and the memory requirement are still on the order of O(N log (N)) and O (N), respectively, as in the SMCG method. Ocean surfaces with size 1024 wavelengths (21.9 m at 14 GHz) and spatial spectrum bandwidth between 0.858 rads/m (corresponding to the longest scale of 341.3 wavelengths) and 4691.5 rads/m (corresponding to the shortest scale of 1/16 wavelengths), which is rather wide to be regarded as a full spectrum, are studied. The maximum of the electromagnetic wavenumber-surface rms height product is up to 25.18. The surface is modeled as a lossy dielectric surface with large relative permittivity rather than as a perfectly conducting surface, which is often adopted as an approximation in the active remote sensing of ocean surfaces. A relatively high sampling density is used to ensure accuracy. The effects of the low and high portions of the spectrum on the emissivity are studied numerically. Monte Carlo simulation for ocean surfaces is also performed by exploiting the efficiency of the multilevel expansion method and the use of parallel computing techniques. The convergence of the results with respect to the sampling density is also illustrated.

Wavelet-based simulations of electromagnetic scattering from large-scale two-dimensional perfectly conducting random rough surfaces

Simulations of electromagnetic waves scattering from two-dimensional perfectly conducting random rough surfaces are performed using the method of moment (MoM) and the electric field integral equation (EFIE). Using wavelets as basis and testing functions, the resulting moment matrix is generally sparse after applying a threshold truncation. This property makes wavelets particularly useful in simulating large-scale problems, in which reducing memory storage requirement and CPU time are crucial. In this paper, scattering from Gaussian conducting rough surfaces of a few hundred square wavelengths are studied numerically using Haar wavelets. A matrix sparsity less than 10% is achieved for a range of root mean square (RMS) height at eight sampling points per linear wavelength. Parallelization of the code is also performed. Simulation results of the bistatic scattering coefficients are presented for different surface RMS heights up to 1 wavelength. Comparisons with sparse-matrix/canonical-grid approach (SM/CG) and triangular discretized (RWG basis) results are made as well. Depolarization effects are examined for both TE and TM incident waves. The relative merits of the SM/CG method and the present method are discussed.

Time-domain Green's functions for the source of HMD in multilayered structures

Planar microwave circuits etched on a dielectric substrate are usually backed by a ground plane. Very often, there are slots on the ground plane for specific coupling purposes. Transient analyses of scattering and radiation from these slots are important problems in understanding the electromagnetic compatibility of the microwave circuits. In modeling the aperture-coupling problem, we replace the electric field across the slots by an equivalent magnetic current. For the transient analyses of the planar layered structures, the time-domain integral equation (TDIE) formulation involves a more efficient discretization compared with the method of finite-difference time-domain (FDTD), where volume discretization is required. Using the TDIE method to analyze the planar structures with slots on the conducting ground plane, the time-domain Greens functions for the magnetic currents in the layered medium are required. Using a numerical method based on the combination of full wave discrete image theory (FWDI) and fast Fourier transform (FFT), Greens functions for an impulsive electric dipole on the top of dielectric substrate with a conducting ground plane were obtained (see Yu, Y. et al., Proc. CJMW, 2000; Xu, Y. et al., Electronics Letters, vol.36, no.22, p.1855-7, 2000). We apply the same algorithm to obtain the time-domain Greens functions for the magnetic source on the conducting ground plane. Numerical results are compared with those of rigorous solutions obtained by the Cagniard-de Hoop method. Good agreements are found between these two methods.

Emissivities and back-scattering coefficients of random lossy dielectric rough surfaces at microwave frequencies based on 3-dimensional numerical simulations

Presents results for full three-dimensional simulations of numerical solutions of Maxwell equations for random rough surfaces. Emissivities and back-scattering coefficients are calculated for soil surfaces and ocean surfaces at microwave frequencies. This is unlike analytical methods or empirical models in that different roughness parameters are used for different microwave frequencies and for active and passive remote sensing. With advances in computational electromagnetics and available computer resources, we calculate 3-dimensional Maxwell equation solutions with a 2-D rough surface for practical remote sensing applications. Highlights of our numerical approach are: (i) the sparse-matrix canonical grid method (SMCG) and the physics-based two-grid method (PBTG) are used. (ii) An algorithm has been implemented for parallel computing. Dense discretization of the surfaces is used because of high permittivity and the energy conservation check is verified. We present results for soil surfaces in L- and C-bands, and for ocean surfaces at 19 GHz.

Simulation of wave scattering from rough surfaces using single integral equation and multilevel sparse-matrix canonical-grid method

There is an interest to simulate electromagnetic scattering from random rough surfaces, which finds applications in remote sensing of ocean, soil and ice. For analysis of a full vector wave scattering from a 2D dielectric random rough surface, two methods have been extensively investigated. One of these is based on the sparse-matrix canonical-grid (SMCG) strategy (see Tsang, L. et al., IEEE Trans. Antenna Propagat., vol.43, p.851-9, 1995; Pak, K. et al., J. Opt. Soc. Amer. A., vol.14, p.1515-29, 1997; Li, Q. et al., IEEE Trans. Antennas Propagat., vol.48, p.1-11, 2000). Traditionally, for simulation of scattering from a dielectric body, coupling equations are employed. Six surface unknowns at each surface point are used by the SMCG method. To reduce the number of surface unknowns and to alleviate the limitation on the surface roughness, a new algorithm based on a single integral equation formulation (see Yeung, M.S., IEEE Trans. Antennas Propagat., vol.47, no.10, p.1615-22, 1999) and a multilevel expansion version of the original SMCG method is developed. We make use of the Rao-Wilton-Glisson (RWG) triangular basis function that can better model the surface than the collocation method. Integration of the RWG basis function and the multilevel sparse-matrix canonical-grid method makes this single integral equation formulation suitable for simulation of scattering from a random surface of arbitrary roughness.

Moment method simulations of electromagnetic scattering from conducting random rough surfaces using wavelet basis

Electromagnetic (EM) scattering from randomly rough surfaces has drawn an increasing attention due to its important applications in remote sensing of the ocean, soil and ice. The proposed wavelet approach is numerically exact, without limitations on the RMS height and slope. The accuracy we want is exclusively controlled by the sparseness of the impedance matrix, which is demonstrated in the order of O(/spl beta/log/sub 2/N/sub x//N/sub x/). This conclusion is drawn based on taking the bistatic scattering coefficient (BSC) predictions as the simulation objectives, and an eight point sampling per linear wavelength is assumed.

Parallel implementation of the sparse-matrix canonical grid method for two-dimensional lossy dielectric random rough surfaces (3D scattering problems) on a Beowulf system

Using a parallel code we compute the bistatic scattering coefficients of a Gaussian random rough surface.