Large-Scale Structures in Acoustics and Electromagnetics Proceedings of a Symposium (1996) / Chapter Skim
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10 AN OVERVIEW OF THE APPLICATION OF THE METHOD OF MOMENTS TO LARGE BODIES IN ELECTROMAGNETICS
10 AN OVERVIEW OF THE APPLICATION OF THE METHOD OF MOMENTS TO LARGE BODIES IN ELECTROMAGNETICS
Pages 204-220
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From page 204... ...
A second advantage of integral equation methods is that the elements in the MM matrix equation are all expressed as integrals of fields, as opposed to derivatives of fields in the differential methods. The averaging process of integration tends to make integral equation methods more stable and reliable than differential equation methods.
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This procedure will reduce the integral equation to a system of N simultaneous linear algebraic equations that can be compactly written as the order N matrix equation [Z]
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CPU time to solve the matrix equation 2 by direct LU decomposition always dominates. The important point is that as the electric size of the body increases, the computer resources increase very rapidly.
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SUMMARY OF METHODS Iterative Solutions Without doubt, the main obstacle to the application of the MM to electrically large bodies is the O(N3) CPU time to solve the matrix equation 2 by direct methods such as LU decomposition.
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V formulation of Figure 10.2(b) , the iterative solution is first solved at the fine level I, then the coarser level 2, and finally the coarsest level 3.
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As illustrated in Figure 10.3, in the FMM the basis fimctions are collected into P groups of p = N/ P basis functions each. If the groups containing the expansion and weighting functions are in the far zone of each other, then the contribution from aZZ of the expansion functions in the expansion group can be evaluated through the use of a multipole expansion that requires far fewer operations than the straightforward superposition indicated in (10.7~.
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The N one body solutions are then combined to form N /2 two body solutions, which are then combined to form N /4 four body solutions, and the process is continued until the Nbody problem is solved. As opposed to the iterative methods described above, these recursive methods determine the exact (meaning the solution that would have been obtained if the N body problem were attacked directly)
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matrix would save storage as well as CPU time by using sparse matrix methods for solving the matrix equation. Further, if the essentially zero elements could be determined a priori, then CPU time could also be saved in the matrix fill by simply not computing the essentially zero elements of Z
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Typically MM solutions employ subsectional basis functions defined over an electrically small region of the body, not exceeding ~ / 4 Th~.~P. Small ~.~1tTPnt~ hemp Pc!
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As the electrical size of the body increases, and the total number of basis functions is increased, the total number of significant matrix elements remains approximately constant. Wavelet Basis Functions Although waveless have many interesting properties that make them useful in signal processing applications and as basis functions for different numerical methods (Daubechies, 1992)
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. An advantage of the Gaussian beam expansion functions is that their fields can be expressed as a summation of analytic terms, thereby avoiding the necessity of a numerical integration.
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, however, now it is an order Nit << N matrix equation. \ \ \ Section 1 , ~ J1 1 ~ ' i' S 1 \ Section 2 Figure 10.6 In the MM/Green's function method, Section 1, but not Section 2, of the body is replaced by free space and equivalent currents.
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The Hybrid GTD/MM Method A virtual axiom in the MM, rooted in the Sampling Theorem, is that one needs at least 4 expansion functions per linear wavelength. Although one can develop "better" basis functions that can produce more accurate results with fewer unknowns per wavelength, in the end the number of unknowns is still proportional to the electrical size of the body (see Table 10.1)
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In the MM regions near the edges, the shape of the current may be complex, and is represented in terms of standard MM expansion functions. Figure 10.7 illustrates a pulse expansion.
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However, in each case, the total CPU time and storage is still proportional to the electrical size of the body, and thus the MM remains a "low-frequency" technique applicable when the body is not electrically too large. Improvements in these techniques and the use of more and more powerful computers will permit the MM to be applied to higher and higher frequencies.
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41, 97-100. Ludwig, A.C., 1986, "A comparison of spherical wave boundary value matching versus integral equation scattering solutions for a perfectly conducting body," IEEE Trans.
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Newhouse, 1975, "A hybrid technique for combining moment methods with the geometrical theory of diffraction," IEEE Trans. Antennas Propag.
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