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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.

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.

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.

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.

A bound for a norm of the difference between the computed and exact solution vectors for static, stationary or quasistationary non‐linear magnetic fields is derived by employing the polarization fixed point iterative method. At each iteration step, the linearized field is computed by using the finite element method. The error introduced in the iterative procedure is controlled by the number of iterations, while the error due to the chosen discretization mesh is evaluated on the basis of the hypercircle principle.

This paper presents an efficient procedure for reconstructing the aged region of ferromagnetic material.

The aging of the ferromagnetic material leads to changes of B ‐H relationship. This property may be used for detection of the degraded parts of ferromagnetic pieces. In numerical procedures the region with a possible aged zone is described by a finite number of subdomains where the flaw vector is defined with binary entries.

Because of the small B ‐H changes, the magnetic field modifications linearly depend (matrix T) by the flaw vectors. Using a double Gauss pivotation scheme, an enough well conditioned and invertible submatrix is extracted from the matrix T. The unknowns associated with this submatrix (called main unknowns) can be easily obtained by a linear relationship from the rest of the unknowns (called minor). In the set of the minor unknowns we search for that vector which gives the smallest error of the principal unknowns in comparison with the values 0 or 1.

This procedure leads to a spectacular increasing of the efficiency in comparison with the known procedures.

The purpose of this paper is to present a simplified rigorous mathematical formulation of the problem of electric currents induced in thin shields with holes yielding more efficient numerical computations with respect to available methods.

A surface integral equation satisfied by the current density was constructed, which is, subsequently, represented at any point by linear combinations of novel vector basis functions only associated with the interior nodes of the discretization mesh, such that the current continuity is everywhere insured. The existence of the holes in the shield is taken into account by associating only one surface vector function with each hole. A method of moments is then applied to compute the scalar coefficients of the vector functions employed.

It was found that the induced current distribution for shields with holes having the complexity of real world structures can be determined with a satisfactory accuracy utilizing a moderate size processor notebook in a time of the order of minutes.

The originality of the proposed method consists in using specialized surface vector functions only associated with individual interior nodes of the shield, its multiply connected structure being efficiently accounted for by introducing one unknown for each hole, instead of unknowns for every node along the hole contours. The method presented is straightforward and highly efficient for mathematical analysis of thin shields with holes, and of other physical fields in the presence of multiply connected surface structures.