Results of Composite Surfaces and Grazing Angle Problems in .NET

Paint qr-codes in .NET Results of Composite Surfaces and Grazing Angle Problems
1.5 Results of Composite Surfaces and Grazing Angle Problems
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Method Gaussian elimination BMIA/CAG
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Table 5.1.3 L = 2500.\, N = 25000. Iteration for BMIA/CAG is based on conjugate gradient and matrix decomposition (Method C).
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Scattering Angle (Degrees)
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Scattering Angle (Degrees)
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Scattering Angle (Degrees)
Scattering Angle (Degrees)
Figure 5.1.5 (a) Close to grazing angle and composite surface. Incident angle (}i = 85 , 9 = 0.125L, b = 400. hI = 0.1A, and h = 0.3.\, h2 = 0.5.\, and 12 = 5-\ for one realization. (b) Close to grazing angle and composite surface. Averaged over 50 realizations. (c) Comparison near-backscattering angle for various surface lengths. Parameters are those of (a). For 2500 wavelengths surface 50 realizations of Fig. 5.1.5b is used and others are averaged over 10 realizations. The bandwidths of L = 200.\ and L = 800.\ are the same as Fig. 5.1.4a, for L = 400.\, b = 300(30.\). (d) Comparison near-backscattering angle with the periodic boundary condition method (PBC). For BMIA/CAG the result of (b) is used. The PBC is for the surface length of 40 wavelengths and averaged over 30 realizations. Parameters are those of (a).
Surface points
Figure 5.1.6 Magnitude of the surface current for one realization for the PBC method with P = 40 wavelengths. Parameters are those of Fig. 5.1.5a.
Physics-Based Two-Grid Method for Lossy Dielectric Surfaces
In the application of the method of moments to the rough surface scattering problem, a common implementation is to use a grid of 10 points per free-space wavelength to discretize the surface. We shall call such a gridding a single coarse grid (SCG). However, in lossy dielectric surfaces, the wavelength in the dielectric medium is much shorter. Thus in scattering by lossy dielectric rough surfaces with high permittivity, there can be rapid spatial variation of surface fields. For microwave remote sensing applications, both wet soil surfaces and ocean surfaces can have large permittivity. Two alternatives were used. The first alternative is to use impedance boundary condition as shown in 4. The disadvantage of this alternative is that an approximation is used in the problem. The second alternative is to use a dense grid with a large number of points (say more than 30 points) per free-space wavelength. We shall call such a gridding a single dense grid (SDG). We have shown in 4 that dense sampling of points is a requirement for energy conservation and an accurate calculation of emissivity. The disadvantage of this second alternative is that there is a large increase in CPU and required memory. The physics-based two-grid (PBTG) method to be discussed in this section is an improvement over these two alternatives in that it has the same accuracy as the single dense grid and yet has a CPU comparable with that of the single coarse grid. To demonstrate the accuracy of the PBTG method, we use it to calculate the emissivity of a random rough surface. In PBTG, two grids were used: a dense grid and a sparse grid. The
2.1 Introduction
sparse grid is that of the usual 10 points per wavelength. The dense grid ranges from 20 to higher number points per wavelength, depending on the relative permittivity of the lossy dielectric medium. The method of PBTG is based on the following two observations: (1) Green's function of the lossy dielectric is attenuative, and (2) Green's function of free space is slowly varying on the dense grid. Because of the Kramer-Kronig relation, a large real part of dielectric constant is associated with a large imaginary part. The first property of the lossy dielectric gives a banded submatrix for the Green's function of the lossy dielectric. When the Green's functions act on the surface field on the dense grid, it corresponds to the product of a sparse matrix with that of a column vector. Thus the convolution of the lossy dielectric Green's function with surface fields is a spatial limited operation. The second property means that the convolution of the free space Green's functions with surface fields on the dense grid is a spatial frequency limited operation. This allows us, when using the free space Green's function to act on the surface fields of the dense grid, to first average the values of surface unknowns on the dense grid and then place them on the coarse grid. PBTG calculates surface field solutions on the dense grid. It needs to be mentioned that PBTG is different from multigrid method [Donohue et al. 1998, Briggs, 1987]. The multigrid method tries to facilitate the convergence of iterations in iterative techniques. It entails discretization of the structure into various grid sizes. The coarse grid corresponds to the low-frequency portion of the solution, while the fine grid corresponds to that of the high-frequency solution. An iterative solution is obtained for each level of discretization, and the solutions are interpolated from the coarse grid to the fine grid. The solution is first obtained in the coarse grid, and then one moves to the next level of fine grid. Once the iterative solution is obtained in the fine grid, one has to go back to the coarse grid to refine the solution. The present method of PBTG, on the other hand, is based on scattering physics. The purpose of PBTG is to speed up the matrix-vector product of two Green's functions convolving with the surface fields on the dense grid. We use two grids in PBTG: a dense grid and a coarse grid. The interaction is divided into (1) a very near field of less than 1 wavelength, (2) a near field of between 1 wavelength and Td wavelengths, and (3) a nonnear-field beyond Td wavelengths. In the numerical simulations performed in this section, T d is fixed at 10 wavelengths. For very near-field interactions, we use a dense grid which is represented by four banded submatrices. For near-field and non-near-field interactions, the free-space Green's function is slowly varying on the dense grid. We average the fields on the dense grid to