Impedance Matrix Localization Research Articles

Decomposing Mom Matrix To Sparse Matrices for Wire Antennas and Wire Scattering By Mei Method

Recently, fast computational methods for directly thinning the matrix of Method of Moment (MoM), such as Impedance Matrix Localization (IML) [1], wavelet expansions [2], and Reduced Expansion and Field Testing (REFT) [3], have been introduced. In this paper, we propose a simple approach to thin the MoM matrix by using the concept of Measured Equation of Invariance (MEI) [4]. This approach is referred to as Matrix Decomposition by MEI (MDMEI). A little effort is required to add the resulting method to any of a variety of MoM programs for solving problems of wire antennas, and wire scattering. When sparsity rate (ratio of non-zero elements to total elements) of the MDMEI matrix reaches to 2.8%-5.6%, the results for a thin wire antenna with 8λ length and a thin wire scattering with 16λ length obtained. by the MDMEI is in excellent agreement to the MoM results.

Impedance matrix localization based fast multipole acceleration

A technique for applying the impedance matrix localization (IML) method in conjunction with the fast multipole method (FMM), termed the hybrid IML‐FMM, is presented. This technique significantly reduces the matrix‐vector multiplication time in the iterative solution of the method of moment equations arising from the integral‐equation‐based analysis of two‐dimensional electromagnetic scattering phenomena. The reduction in computational time is accomplished through an IML‐based sparsification of the FMM far‐field transformation matrices. Numerical experiments illustrate the effect of various IML‐FMM truncation parameters on the sparsity obtainable, the corresponding reduction in matrix‐vector multiplication time, and the solution accuracy.

Solution of impedance matrix localization form of moment method problems in five iterations

The impedance matrix localization (IML) method replaces the usual full N × N moment method matrix by a sparse N × N matrix. Typically, this matrix has about 50N nonzero elements. Iterative methods require one (or a few) matrix vector product(s) at each iteration. Each product can be accomplished in 50N operations rather than N2 for the usual moment method. There is a correspondence between the clumps of large numbers in the IML matrix and terms with a ray description such as in geometrical optics and the geometrical theory of diffraction. It is known that using these methods approximate solutions are possible while taking into account only a limited number of multiple interactions. Using this analogy an approximate factored form of the IML matrix is generated by explicitly using only a limited number of direct interactions. However, all multiple interactions involving these direct interactions are (implicitly) taken into account. This effectively gives an approximate inverse which is used in an iterative approach. A reduction in the residual of orders of magnitude per iteration is achieved. The result is a more accurate solution than was possible before and which requires only a few iterations. This method is generally applicable since the factored form is based on the structure of the IML matrix and not on any specific geometry.

Fast integral equation solutions using geometrical‐theory‐of‐diffraction‐like matrices

Frequency domain integral equations allow high‐accuracy direct solvers, while often requiring excessive computational power for large problems. This contrasts with high‐frequency methods such as the geometrical theory of diffraction (GTD), which allow efficient solution of large problems, though often with less accuracy. Use of a special type of expansion functions in integral equations results in the impedance matrix localization method, which localizes the information in the impedance matrix to clumps of large numbers that are analogous to the terms of GTD. Fast numerical methods are developed for calculating these large matrix elements. Then the analogy to GTD is strengthened by developing an integral equation based on sources which radiate only exterior to a closed body. This is a new kind of combined source equation. The peaks (i.e., clumps of large numbers) are now restricted to near the diagonal for a convex body. This both allows a faster matrix solution and further strengthens the analogy to GTD. The local structure of the resulting matrix, its inverse, and the method of finding the inverse are found to be closely analogous to the associated high‐frequency method quantities. This suggests the possibility of deriving numerical GTD coefficients.

A sparse boundary‐element method for scattering from a rigid body of revolution

A sparse boundary‐element method is used to calculate the scattered acoustic field of a rigid body of revolution. The N×N full matrix resulting from the discretization of the Helmholtz integral is transformed to a sparse N×N matrix using the impedance matrix localization (IML) method. When applied to integral equations for wave phenomena, the IML method produces a sparse matrix by partitioning the scatterer into several regions, over which the response is represented by basis functions with directional radiation patterns. The number of important physical interactions is reduced to a small number over each region, leaving a large number of matrix elements with relative magnitudes less than 10−4 which can be approximated by zero. Used in conjunction with existing BEM codes, the IML method reduces the memory and storage requirements from N2 to approximately 100N, while the solution time is O(N) compared to O(N3) for the full untransformed case. The resulting sparse matrix can be approximately inverted leading to a convergent solution in less than five iterations. Results are given for scattering from a hemispherically endcapped cylinder of L/a=110 for ka=1 to ka=17.

Improved impedance matrix localization method (EM problems)

Moment method calculations have the well-known limitations of requiring excessive storage and execution times for even modestly large electromagnetics problems. The impedance matrix localization (IML) method was introduced as a modification to standard moment method calculations to ease these limitations. It utilizes a matrix transformation which effectively changes the basis (testing) functions into ones resembling traveling waves. An improved method that uses an orthogonal transformation to generate standing-wave-like basis functions is presented here. Remarkable improvements are achieved in the numerical stability of the method and in its compatibility with iterative solvers. Furthermore, the correspondence of the large elements in this matrix to geometrical theory of diffraction (GTD) terms is strengthened, as is the possibility of further increasing the speed of iterative solutions by constructing preconditioners based on the pattern of nonzero matrix elements.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Preconditioned iterative solution of IML/moment method problems

The impedance matrix localization (IML) method replaces the usual method of moments matrix Z by a sparse matrix T. For example, when both Z and T are N*N, T generally has about 50 N nonzero elements. Although this allows each iteration of an iterative method to take only 50 N operations (rather than N/sup 2/), the number of iterations still must be decreased to truly have a fast method. For example, often more than N iterations are necessary for standard methods. Several standard and fast iterative methods are compared. These methods all converge to the exact solution of the matrix equation involving T, and the fast ones do so by using an approximate solution derived from a sparse, approximate factorization of T. The approximate factorization is accurate enough to allow a solution for general problems in five iterations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The impedance matrix localization (IML) method for moment-method calculations

The impedance matrix localization (IML) method, a modification of the standard method of moments that can be implemented as a modification to existing computer programs, is examined. This modification greatly eases the excessive storage requirements and long computation times of moment-method approaches by using novel bases and testing function that localize the important interactions to only a small number of elements within the impedance matrix elements can be made so small (typically 10/sup -4/ to 10/sup -6/ in relative magnitude) that they may be approximated by zero. In the case of a two-dimensional body with unknowns on its surface, both analytical arguments and numerical calculations suggest that, for an N*N matrix, about 100N matrix elements will need to be kept, even for very large N. The resulting sparse matrix requires storage for only 100N complex numbers rather than for N/sup 2/ numbers. Similar results are expected in three dimensions. The structure of the resulting matrix problem allows the use of highly efficient solution methods. Results are given for one such possibility: iteration preconditioned by incomplete LU decomposition. >