Convergence acceleration in the polarization method for nonlinear periodic fields

Abstract

Purpose -The purpose of this paper is to present three novel techniques aimed at increasing the efficiency of the polarization fixed point method for the solution of nonlinear periodic field problems. Design/methodology/approach -Firstly, the characteristic B-M resulting from the constitutive relation B-H is replaced by a relation between the components of the harmonics of the vectors B and M. Secondly, a dynamic overrelaxation method is implemented for the convergence acceleration of the iterative process involved. Thirdly, a modified dynamic overrelaxation method is proposed, where only the relation B-M between the magnitudes of the field vectors is used. Findings -By approximating the actual characteristic B-M by the relation between the components of the harmonics of the vectors B and M, the amount of computation required for the field analysis is substantially reduced. The rate of convergence of the iterative process is increased by implementing the proposed dynamic overrelaxation technique, with the convergence being further accelerated by applying the modified dynamic overrelaxation presented. The memory space is also well reduced with respect to existent methods and accurate results for nonlinear fields in a real world structure are obtained utilizing a normal size processor notebook in a time of about one-half of one minute when no induced currents are considered and of about one minute when eddy currents induced in solid ferromagnetic parts are also fully analyzed. Originality/value -The originality of the novel techniques presented in the paper consists in the drastic approximations proposed for the material characteristics of the nonlinear ferromagnetic media in the analysis of periodic electromagnetic fields. These techniques are highly efficient and yield accurate numerical results.

Anderson acceleration for electromagnetic nonlinear problems

“…A similar approach based on the optimal choice of m FP has been studied in Koczka et al (2009) and Außerhofer et al (2008), but in these approaches the stiffness matrix has to be updated at each nonlinear iteration thus the complexity would be the same of NR with a linear or super-linear convergence still lower than quadratic. Hantila et al (2000) and Ciric et al (2011) suggest to adopt a relaxation scheme…”