In this study, we demonstrate an acceleration of flexible generalized minimal residual algorithm (FGMRES) implemented with the method of moments and the fast multipole method (FMM), based on a combined tangential formulation. For the implementation of the FGMRES incorporated with the FMM concept, we propose a new definition of the truncation number for the FMM operator within the inner solver. The proposed truncation number provides an optimal variable preconditioner by controlling the accuracy and computational cost of the inner iteration. Moreover, to further accelerate the convergence, we introduce the concept of a multistage preconditioner. Numerical experiments reveal that our new version of FGMRES, based on the proposed truncation number for the inner solver and the multistage preconditioner, achieves outstanding acceleration of the convergence for large-scale and practical electromagnetic scattering and radiation problems with several levels of geometrical complexity.

In this paper, the performance of the induced dimension reduction (IDR) method implemented along with the method of moments (MoM) is described. The MoM is based on a combined field integral equation for solving large-scale electromagnetic scattering problems involving conducting objects. The IDR method is one of Krylov subspace methods. This method was initially developed by Peter Sonneveld in 1979; it was subsequently generalized to the IDR(s) method. The method has recently attracted considerable attention in the field of computational physics. However, the performance of the IDR(s) has hardly been studied or practiced for electromagnetic wave problems. In this study, the performance of the IDR(s) is investigated and clarified by comparing the convergence property and memory requirement of the IDR(s) with those of other representative Krylov solvers such as biconjugate gradient (BiCG) methods and generalized minimal residual algorithm (GMRES). Numerical experiments reveal that the characteristics of the IDR(s) against the parameter s strongly depend on the geometry of the problem; in a problem with a complex geometry, s should be set to an adequately small value in order to avoid the "spurious convergence" which is a problem that the IDR(s) inherently holds. As for the convergence behavior, we observe that the IDR(s) has a better convergence ability than GPBiCG and GMRES(m) in a variety of problems with different complexities. Furthermore, we also confirm the IDR(s)'s inherent advantage in terms of the memory requirements over GMRES(m).

This paper presents flexible inner-outer Krylov subspace methods, which are implemented using the fast multipole method (FMM) for solving scattering problems with mixed dielectric and conducting object. The flexible Krylov subspace methods refer to a class of methods that accept variable preconditioning. To obtain the maximum efficiency of the inner-outer methods, it is desirable to compute the inner iterations with the least possible effort. Hence, generally, inaccurate matrix-vector multiplication (MVM) is performed in the inner solver within a short computation time. This is realized by using a particular feature of the multipole techniques. The accuracy and computational cost of the FMM can be controlled by appropriately selecting the truncation number, which indicates the number of multipoles used to express far-field interactions. On the basis of the abovementioned fact, we construct a less-accurate but much cheaper version of the FMM by intentionally setting the truncation number to a sufficiently low value, and then use it for the computation of inaccurate MVM in the inner solver. However, there exists no definite rule for determining the suitable level of accuracy for the FMM within the inner solver. The main focus of this study is to clarify the relationship between the overall efficiency of the flexible inner-outer Krylov solver and the accuracy of the FMM within the inner solver. Numerical experiments reveal that there exits an optimal accuracy level for the FMM within the inner solver, and that a moderately accurate FMM operator serves as the optimal preconditioner.

A compact representation of the Green function is proposed by applying the discrete wavelet concept in the k-domain, which can be used for the acceleration of scattered field calculations in integral equation methods. A mathematical expression of the Green function based on the discrete wavelet concept is derived and its characteristics are discussed.

We calculated the shielding characteristic of a three-dimensional array of strip conductors by using the electric field integral equation method and its expansion to an array structure. From reflection coefficients, the effective permittivity of the array is calculated. The effective permittivity becomes negative in the frequency range above resonance, in which the electromagnetic waves travel through the material in an evanescent mode and the transmission coefficient becomes very small.

The electric currents on the upper, lower and side surfaces of the patch conductor in a circular microstrip antenna are calculated by using the integral equation method and the characteristic between the electric currents on the upper and lower surfaces is compared. The integral equation is derived from the boundary condition that the tangential component of the total electric field due to the electric currents on the upper, lower and side surfaces of the patch conductor vanishes on the upper, lower and side surfaces of the patch conductor. The electric fields are derived by using Green's functions in a layered medium due to a horizontal and a vertical electric dipole on those surfaces. The result of numerical calculation shows that the electric current on the lower surface is much bigger than that on the upper surface and the input impedance of microstrip antenna depends on the electric current on the lower surface.

Integral equation method is used to compute three-dimension-structure capacitance in this paper. Since some multi-conductor structures present regular periodic property, the periodic cell is used to reduce the computational domain with adding appropriate magnetic and electric walls. The periodic Green's function in the integral equation method is represented in the form of infinite series with slow convergence. In this paper, Shanks transformation is used to accelerate the convergence. Numerical examples show that the proposed method is accurate with a much higher efficiency in capacitance extraction for 3-D periodic structures.

In this paper, we present an analysis of the microstrip lines whose strip conductors are of various cross-sections, such as rectangular cross-section, triangle cross-section, and half-cycle cross-section. The method employed is the boundary integral equation method (BIEM). Numerical results for these microstrip lines demonstrate various shape effects of the strip conductor on the characteristics of lines. The processing technique on the convergence of the Green's function is also described.