Harmonic balance (HB) is well established as the principal numerical technique for the simulation of microwave circuits operating in strongly nonlinear conditions. However, very large problems cannot be tackled by ordinary HB, because the necessary computer resources may exceed the capabilities of even the largest computer systems, due to the memory storage and CPU time requirements that are proportional to the square/cube, of the total number of unknowns. Unfortunately, in the circuit-level analysis of nonlinear circuits and subsystems (up to entire front-ends) for telecommunications systems, numerical problems of many million unknowns may be encountered as a matter of course. This is due to the fact that modern integrated subsystems normally contain several tens up to several hundreds of nonlinear devices, and at the same time must process digitally modulated signals adopting complex modulation formats. In turn, an adequate description of such signals requires high spectral resolutions, and thus spectra of several tens of thousands of discrete lines.

In order to cope with these difficulties, and, at the same time to preserve the desired features of harmonic balance, an integrated set of new HB methods have been devised. These innovative methods rely upon inexact Newton and Krylov subspaces methods. The HB equations are solved by GMRES-Newton algorithm and the storage of the Jacobian matrix is substituted by a series of matrix-vector products. The products are realized in an efficient way by an original technique that avoids to assemble the matrices that form the Jacobian matrix and uses the structure and properties of each matrix. Ad hoc preconditioners have been devised, principally based on incomplete factorizations, and the number of iterations in each linear solution were drastically reduced.

Photonic Crystal Fibers

Nowadays many research groups are working on a particular type of optical fiber known as Photonic Crystal Fibers, PCFs, because of their periodical structure similar to the crystals that constitute the external part of the kernel. The reason of such an interest is in the particular characteristic of PCFs. For instance, the trend of the dispersion characteristic is strictly related to their structure that can be so designed according to specific necessity; on the other hand, they can work in mono-modality for the wave length or spectral zone where traditional fibers are instead multi-modal.

In order to find the configurations
that exploit in a very efficient way these and other interesting properties of
PCfs, it is necessary to have general models and efficient numerical algorithms
to simulate very large problems.

Finite Element Methods seems to be the most appropriates methodologies to
study PCFs.

The research will try to generalize the existing models to consider
new complex structure in order to take into account small dimension
unhomogeneity in their transverse section.

The resolution of the electromagnetic problem to determine the characteristic of the field guided by PCFs leds to the solution of a huge dimension generalized eigenvalue problem. Moreover, to take into account the loss of field irradiation, in the generalized eigenvalue problem we have to consider non hermitian complex matrices.

The researches are
done in collaborations with the Department of

Electronics, Computer Science and System (DEIS) –

University of Bologna