Junsheng Zhao

Fast inhomogeneous plane wave algorithm for the fast analysis of two-dimensional scattering problems

A novel algorithm, the fast inhomogeneous plane wave algorithm (FIPWA), has been developed to accelerate the solution of integral equations pertinent to the analysis of the scattering from two-dimensional perfect electric conducting surfaces. Unlike the fast steepest descent path algorithm, the proposed technique directly interpolates the far-field pattern of the source group and matches it along a modified steepest descent path. A novel approach, which results in a diagonal translator with built-in interpolation coefficients, is proposed. The computational complexity per matrix-vector multiplication of a two-level implementation of the proposed FIPWA is O(N4/3) and the multilevel implementation further reduces the complexity to O(NlogN), where N is the number of unknowns in the discretized integral equation. It is shown that this technique outperforms the previously developed fast methods such as the fast mulitpole method and the ray-propagation fast multipole algorithm.

A surface integral equation formulation for low-frequency scattering from a composite object

A surface integral equation formulation is derived for low-frequency scattering from a composite object. This formulation allows those algorithms for finding the loop-tree basis on simple surfaces to be applied to the complicated interfaces of a composite object. By including the residue term in the K operator, the present formulation induces the interface boundary conditions automatically, and the resultant matrix equation can be solved directly without using the O(N/sup 2/) number-of-unknowns-reduction scheme. Numerical results validate that the algorithm is stable even at extremely low frequencies and for closely spaced structures.

Generalized PMCHWT formulation for low-frequency multi-region problems

The loop-tree basis is used for solving low-frequency scattering problems by the moment method, since the traditional RWG basis will lead to an ill-conditioned impedance matrix when the frequency is low. However, when a scatterer comprises several regions of different materials, the interfaces between different regions form a complicated surface. It is difficult to find the loop-tree basis on such an arbitrary complicated surface. [I] presented a method, which we shall call contact-region modeling (CRM), to treat this problem. Then a number-of-unknown-reduction (NOUR) scheme was used to impose the boundary conditions as well as to reduce the redundant unknowns. However, the O(N2) computational complexity of the NOUR scheme hinders the scattering problem from being solved by some fast solvers, for instance, the multilevel fast multipole algorithm(MLFMA). In [7] and [8 ] , it was pointed out that, in general, the K operator should include both the residue term and the Cauchy principal value integral. In the present work, by intepreting the K operator in this way in the PMCHWT (traditonally named as PMCHW, but we follow (21 here) equations obtained by applying CRM, the boundary conditions can be automatically induced when different regions touch each other and the corresponding matrix equation can be solved directly without using the NOUR scheme. Thus the formulation is ready for a fast solver. Another important issue for multi-region problems is the near-patch integration. By employing Graglias method (41 to integrate the singular term analytically, stable results are obtained for any distance, from non-zero separation to zero separation, between different regions. Chen et al.

Approximate inverse preconditioner for near resonant scattering problems

To solve the method of moments (MoM) matrix calculation, iterative methods are becoming popular as a solution technique for such problems. These methods require one or two matrix-vector multiplications per iteration. To speed up the matrix-vector multiplications in the iterative solvers, the fast multipole method (FMM) and its multilevel extension, multilevel fast multipole algorithm (MLFMA) are developed to obtain fast solution for scattering problems. But they are more advantageous if the number of iterations for convergence is much smaller than the number of unknowns. However, for strongly resonant problems and/or strong multiple reflection problems, these methods require a very large number of iterations for convergence. In this paper, to account for these near interactions, the approximate inverse of the block banded coefficient matrix is introduced as a preconditioner. By means of the matrix partitioning method, the computing time required to set up this preconditioner is O(N), which is indispensable for applying to O(Nlog N) fast algorithms.

Fast multilevel techniques for solving integral equations in electromagnetics

In this paper, we describe several fast methods for solving electromagnetic scattering problems. Notably, we discuss iterative surface integral equation solvers such as the 3D fast multipole method, multilevel fast multipole method, rough surface scattering using steepest descent path fast multipole method, and the thin-stratified media fast multipole method.

Solution of large differential signal problem using two different integral equation based methods

The ability to solve large, realistic EM problems is becoming very important with the increase in the clock and radio frequencies in VLSI based systems. Fortunately, a similar increase in the ability to solve such problems has been accomplished with the advent of several new solution approaches. In this paper, the authors compare the results of two solution approaches which are based on an integral equation formulation of Maxwells equation. The formulation and solution approaches are very different so that the comparison helps to establish confidence in the validity of the results. The problem solved consists of a printed circuit board with a differential mode driver and a differential wire which extends 1 m beyond the board. The problem is solved on conventional high end server workstations.

Fast algorithms for the electromagnetic simulation of planar structures

This paper reviews the state of the art in fast integral equation techniques for solving large scale electromagnetic radiation and compatibility problems. The fast multipole method (FMM) and its frequency and time domain derivatives are discussed. These techniques permit the rapid evaluation of fields due to known sources and hence accelerate the solution of boundary value problems arising in the analysis of a wide variety of electromagnetic phenomena. Specifically, the application of the steepest descent fast multipole method (SDFMM) and the thin stratified medium fast multipole algorithm (TSM-FMA) to the frequency domain analysis of radiation from microstrip structures residing on finite and infinite substrates and ground planes, respectively, is described. In addition, the extension of the FMM concept to plane wave time domain (PWTD) algorithms that permit the analysis of transient phenomena is outlined.

An improved fast steepest descent path algorithm

We have presented a modification of the fast steepest descent gradient path algorithm (FASDPA) for 2-D scattering. The computational cost of the FASDPA is lower than that of the previously developed fast multipole method (FMM) and ray propagation fast multipole algorithm (RPFMA), both for two-level and multi-level implementations. The method, which uses an interpolated far field pattern to arrive at an efficient representation of the integral along the steepest descent path, has inherent advantages such as simplicity and ease of diagonalization.

MLFMA for solving boundary integral equations of 2D electromagnetic scattering at low frequencies

A normalized 2D MLFMA (multilevel fast multipole algorithm) with a computational complexity of O(N) for low frequencies is developed. This normalized 2D MLFMA can be used independently not only at low frequencies, but be merged easily with the dynamic algorithm to solve large-scale problems with dense subgridded areas. Some numerical computations are performed, and good agreement is observed with the plain conjugate gradient method. The null space problem for boundary integral equations at low frequencies is also studied.

Convergence improvement for iterative solutions of the electric fields integral equation at very low frequencies

The matrix equations of the electric field integral equation (EFIE) at very low frequencies based on the loop-tree basis and the loop-star basis are transformed by a connection matrix which is a rearrangement of the basis. The new representation of the matrix equation can be solved by iterative solvers. It converges fast and no low frequency break-down occurs in the numerical computation. A method to perform the multiplication of the inverses of the connection matrix and its transpose with a vector for the tree basis with only O(N) floating-point operations is also developed. This work is incorporated into the low frequency multilevel fast multipole algorithm (LF-MLFMA) to solve large problems all the way from zero frequency to electrodynamic frequencies. The matrix transformation does not increase the computational complexity. The memory requirements and the number of floating-point operations still scale as O(N).