The multigrid preconditioned flexible GMRES solver for hierarchical TVFEM analysis

Abstract

mm, length l ϭ 23.31 mm, substrate height b ϭ 0.254 mm, cylinder spacing W ϭ 1.2 mm, permittivity r ϭ 2.2, and loss tangent tan ␦ ϭ 0.0009; at K band, SIW width a ϭ 6 mm, length l ϭ 8 mm, substrate height b ϭ 0.254 mm, cylinder spacing W ϭ 1 mm, permittivity r ϭ 2.2, and loss tangent tan ␦ ϭ 0.0009.In Figures 5 and 6, we can see that, the difference between calculation and simulation results increase with the cylinder radius and frequency band. However, the deviation is not big even at K-band. At X-band, the maximum deviation is about 1.84%, while it is 2.98% at K-band. The results of calculation and simulation imply the validity and accuracy of our derived formula; all the deviations are less than 3%. There are three potential reasons causing the errors. First, the equations of electromagnetic fields in SIW resonance cavity is not the same as those in RW resonance cavity strictly; second, the current density distribution on metal cylinders change along the radius of cylinders, which definitely have an impact on integral; third, the simulation is implemented by commercial software HFSS, error of the software is also a factor potentially. CONCLUSIONSThe quality factor of the SIW resonance cavity is investigated in this paper. The conductor losses on the sidewalls, front/end walls and top/bottom planes of the cavity have been derived separately, the formula of the quality factor for the SIW cavity is proposed for the first time. The calculations from the derived formula at several frequency bands have been made, good agreements are observed between the calculations and the numerical HFSS simulations. The maximum deviations at S-, X-, and K-bands are 1.52%, 1.84%, and 2.98%, respectively, all less than 3%, demonstrating the validity and the accuracy of our formula. In addition, the error factors are discussed also in the paper. ACKNOWLEDGMENTThe authors express their gratitude to the financial support of the National Science Foundation of China under Grant 60471025 and the Natural Science Foundation of Jiangsu Province under Grant BK2004135. Poling optical fiber to induce a second-order optical nonlinearity allows for fiber-based devices, such as electro-optic modulators, optical switches, and frequency converters, and increased opportunity for more fundamental research into second-order nonlinearity in macroscopic centrosymmetric crystal. Poled silica fiber devices offer low coupling losses and long interaction lengths. Thermally poling is probably the most popular and promising among all poling techniques. Much higher electric fields can be applied by including internal electrodes than is possible with external electrodes because of the high dielectric breakdown of silica compared with that of air. The maximum electro-optic coefficient of 6 pm/V in silica is observed in a fiber with internal electrodes [1]. Internal electrodes are always achieved by inserting thin metal wires into fibers. However, the lengths of electrodes are usually not more than 10 cm. Manufacturing longer fibers for poling is desirable becaus...

A novel flexible GMRES with deflated restarted for efficient solution of electromagnetic problems

“…The inner iteration operation can be considered as a preconditioner of outer iteration. FGMRES is very effective for some EM Scattering Problems [13, 14].…”