×

Multiscalet basis in Galerkin’s method for solving three-dimensional electromagnetic integral equations. (English) Zbl 1154.78003

The author shows how multiscalets, which are building blocks for multiwavelets, can be used to solve the electromagnetic integral equations electric field integral equation and magnetic field integral equation. Two geometries which both can be parametrized by one patch are considered: the sphere and a rough surface where \(z=f(x,y)\), \(f\) is a random function and \((x,y)\) belongs to the unit square. The author emphasizes that the Sobolev orthogonality property of multiscalets implies that the developed Galerkin method, with multiscalets as basis and test functions, behaves as a collocation method.
For the sphere case the standard spherical coordinates are used. In both cases a uniform Cartesian mesh of the unit square are used as collocation points and the discretized integral equation together with directional derivatives of the equation are solved in a least square sense (since the number of unknowns are less than the number of equations). On the particular test cases it is shown that the method can produced more accurate results to a lower cost than traditional methods but there is no indication how more complicated geometries can be handled.

MSC:

78A45 Diffraction, scattering
65T60 Numerical methods for wavelets
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Pan, Wavelets in Electromagnetics and Device Modeling (2003)
[2] Beylkin, Fast wavelet transforms and numerical algorithms I, Communications on Pure and Applied Mathematics 44 (2) pp 141– (1991) · Zbl 0722.65022
[3] Steinberg, On the use of wavelet expansions in the method of moments, IEEE Transactions on Antennas and Propagation 41 (5) pp 610– (1993)
[4] Sabetfakhri, Analysis of integrated millimeter-wave and submillimeter-wave waveguides using orthonormal wavelet expansions, IEEE Transactions on Microwave Theory and Techniques 42 (12) pp 2412– (1994)
[5] Wagner, A study of wavelets for the solution of electromagnetic integral equation, IEEE Transactions on Antennas and Propagation 43 (8) pp 802– (1995)
[6] Deng, Fast solution of electromagnetic integral equations using adaptive wavelet packet transform, IEEE Transactions on Antennas and Propagation 47 (4) pp 674– (1999) · Zbl 0949.78022
[7] Zunoubi, A combined Bi-Cgstab (1) and wavelet transform method for EM problems using method of moments, Progress in Electromagnetics Research pp 205– (2005)
[8] Pan, Multiwavelet based moment method under discrete Sobolev-type norm, Microwave and Optical Technology Letters 40 (1) pp 47– (2004)
[9] Tong, Full-wave analysis of coupled lossy transmission lines using multiwavelet-based method of moments, IEEE Transactions on Microwave Theory and Techniques 53 (7) pp 2362– (2005)
[10] Robinson, Least Squares Regression Analysis in Terms of Linear Algebra (1981) · Zbl 0525.62059
[11] Longley, Least Squares Computations Using Orthogonalization Methods (1984)
[12] Björck, Numerical Methods for Least Squares Problems (1996)
[13] Ishimaru, Wave Propagation and Scattering in Random Media 2 (1978)
[14] Tsang, Theory of Microwave Remote Sensing (1985)
[15] Beckmann, The Scattering of Electromagnetic Waves from Rough Surfaces (1987)
[16] Tsang, Scattering of Electromagnetic Waves: Numerical Simulations 2 (2001)
[17] Pak, Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces based on Monte Carlo simulations, Journal of the Optical Society of America A 12 (11) pp 2491– (1995)
[18] Johnson, Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: a comparison of Monte Carlo simulations with experimental data, IEEE Transactions on Antennas and Propagation 44 (5) pp 748– (1996)
[19] Wagner, Monte Carlo simulation of electromagnetic scattering from two-dimensional random rough surfaces, IEEE Transactions on Antennas and Propagation 45 (2) pp 235– (1997)
[20] Xia, An efficient algorithm for electromagnetic scattering from rough surface using a single integral equation and the sparse-matrix canonical grid method, IEEE Transactions on Antennas and Propagation 51 (6) pp 1142– (2003)
[21] Kuo, Scattering from multilayer rough surfaces based on the extended boundary condition method and truncated singular value decomposition, IEEE Transactions on Antennas and Propagation 54 (10) pp 2917– (2006)
[22] Zahn, Numerical simulation of scattering from rough surfaces: a wavelet-based approach, IEEE Transactions on Antennas and Propagation 48 (2) pp 246– (2000)
[23] Poggio, Computer Techniques for Electromagnetics (1973)
[24] Balanis, Advanced Engineering Electromagnetics (1989)
[25] Li, Numerical simulation of conical diffraction of tapered electromagnetic waves from random rough surfaces and applications to passive remote sensing, Radio Science 29 (3) pp 587– (1994)
[26] Braunisch, Tapered wave with dominant polarization state for all angles of incidence, IEEE Transactions on Antennas and Propagation 48 (7) pp 1086– (2000) · Zbl 1368.78035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.