Search results

The aim of this paper is to accelerate the convergence of iterative methods on ill‐conditioned linear systems of equations.

First a brief numerical analysis is given of left preconditioners on ill‐conditioned linear systems of equations. From this result, it is deduced that a double preconditioning approach may be better. Then, a double preconditioner based on an iterative diagonal scaling method and an incomplete factorization method is proposed. The efficiency of this approach is illustrated on two finite element models produced by computational electromagnetism.

The double preconditioning approach is efficient for 2D and 3D finite element problems. The bi‐conjugate gradient algorithm always converges when it is double preconditioned. This is not the case when a simple incomplete factorization method is applied. Furthermore, when the two preconditioning techniques lead to the convergence of the iterative solving method, the double preconditioner significantly reduces the number of iterations in comparison with the simple preconditioner. On the proposed 2D problem, the speed‐up is between 6 and 32. On the proposed 3D problem, the speed‐up is between 13 and 20. Finally, the approach seems to reduce the growth of the condition number when higher‐order finite elements are used.

The paper proposes a particular double preconditioning approach which can be applied to any invertible linear system of equations. A numerical evaluation on a singular linear system is also provided but no proof or analysis of stability is given for this case.

The paper presents a new preconditioning technique based on the combination of two very simple and elementary methods: a diagonal scaling method and an incomplete factorization process. Acceleration obtained from this approach is quite impressive.

This paper presents the modeling of a current transformer by various methods with the FLUX3D software. The technique used is based on the finite element method coupled with electric circuits. A magnetic scalar potential reduced versus T₀ formulation (T₀ϕ−ϕ) taking into account the electric circuits with an air‐gap is used for this purpose. The air‐gap is described either by a thin volume region or by a surface region.

To model magneto‐harmonic devices including solid conductors with holes when the skin depth is very small.

The 3D finite element magnetic scalar potential formulation combined with the surface impedance condition approximation is used. It allows the modelling of thin skin depth effect at low cost.

The paper shows how to use surface impedance condition for solid conductors with holes, when using the magnetic scalar potential. Specific equations must be added to respect Ampere's theorem. The paper establishes these equations and the coupling with the finite element formulation. The final system of equations is symmetric.

The formulation allows to treat linear material in the magneto‐harmonic assumption.

The use of surface impedance condition with the 3D finite element magnetic scalar potential formulation is well known. The originality is to take into account holes (multiply connected conductors).