Analysis of the Propagation in High-Speed Interconnects for MIMICs by Means of the Method of Analytical Preconditioning: A New Highly Efficient Evaluation of the Coefficient Matrix

The method of analytical preconditioning combines the discretization and the analytical regularization of a singular integral equation in a single step. In a recent paper by the author, such a method has been applied to a spectral domain integral equation formulation devised to analyze the propagation in polygonal cross-section microstrip lines, which are widely used as high-speed interconnects in monolithic microwave and millimeter waves integrated circuits. By choosing analytically Fourier transformable expansion functions reconstructing the behavior of the fields on the wedges, fast convergence is achieved, and the convolution integrals are expressed in closed form. However, the coefficient matrix elements are one-dimensional improper integrals of oscillating and, in the worst cases, slowly decaying functions. In this paper, a novel technique for the efficient evaluation of such kind of integrals is proposed. By means of a procedure based on Cauchy integral theorem, the general coefficient matrix element is written as a linear combination of fast converging integrals. As shown in the numerical results section, the proposed technique always outperforms the analytical asymptotic acceleration technique, especially when highly accurate solutions are required.

Variations on the Brouwer Fixed Point Theorem: A Survey

This paper surveys some recent simple proofs of various fixed point and existence theorems for continuous mappings in R n . The main tools are basic facts of the exterior calculus and the use of retractions. The special case of holomorphic functions is considered, based only on the Cauchy integral theorem.

Efficient Evaluation of Slowly Converging Integrals Arising from MAP Application to a Spectral-Domain Integral Equation

In this paper, we devised an analytical technique to efficiently evaluate the improper integrals of oscillating and slowly decaying functions arising from the application of the method of analytical preconditioning (MAP) to a spectral-domain integral equation. The reasoning behind the method’s application may consistently remain the same, but such a procedure can significantly differ from problem to problem. An exhaustive and understandable description of such a technique is provided in this paper, where we applied MAP for the first time to analysis of electromagnetic scattering from a zero-thickness perfectly electrically conducting (PEC) disk in a planarly layered medium. Our problem was formulated in the vector Hankel transform domain and discretized via the Galerkin method, with expansion functions reconstructing the physical behavior of the surface current density. This ensured fast convergence in terms of the truncation order, but involved numerical evaluation of slowly converging integrals to fill in the coefficient matrix. To overcome this problem, appropriate contributions were pulled out of the kernels of the integrals, which led to integrands transforming into exponentially decaying functions. Subsequently, integrals of the extracted contributions were expressed as linear combinations of fast-converging integrals via the Cauchy integral theorem. As shown in the numerical results section, the proposed technique drastically outperformed the classical analytical asymptotic-acceleration technique.

Curvilinear Integral Theorem for G-Monogenic Mappings in the Algebra of Complex Quaternion

For G-monogenic mappings taking values in the algebra of complex quaternion we prove a curvilinear analogue of the Cauchy integral theorem in the case where a curve of integration lies on the boundary of a domain of G-monogeneity.