Fixed‐point technique in computing nonlinear eddy current problems

Abstract

Eddy currents are investigated in a ferromagnetic bar exposed to a transverse magnetic field. Such an open boundary field problem is analysed applying the Galerkin finite element method coupled with a separation of variables. A steady state is considered, introducing time periodic conditions. The resultant system of nonlinear equations is solved using an iterative procedure based on Brouwer's fixed-point theorem referred to the nonlinear material reluctivity. Numerical results are presented for a massive conductor made of cast steel and cast iron. The eddy-current distribution and characteristics of power losses are illustrated in a graphic form.

Steady-state time-periodic finite element analysis of a brushless DC motor drive considering motion

“…Besides lack of symmetry of the Jacobian matrix [9][10][11][12][13], due to application of the Lagrange multipliers, it becomes an indefinite matrix. A crucial problem in making this model competitive to the transient one (with respect to the execution time), is a choice of an efficient solver of this system of linear equations.…”

“…Because the value of the moment of inertia in physical motors is usually a small number, the updates of T r in consecutive iterations are large and usually nonmonotone. This one, in addition to the known features of the Newton-Raphson method when applied to the time-periodic models regarding a problematic convergence [13], makes the iterative process unstable. For that reason, in the solution of the current problem the Newton-Raphson method will be replaced by the fixed-point technique [13,14].…”

“…This one, in addition to the known features of the Newton-Raphson method when applied to the time-periodic models regarding a problematic convergence [13], makes the iterative process unstable. For that reason, in the solution of the current problem the Newton-Raphson method will be replaced by the fixed-point technique [13,14]. The discretized physical equation of mechanical balance (12) will be substituted by the artificial equation of the form:…”

“…and D Fp is the fixed-point coefficient being constant during the iterative process [13,14]. By proper selection of D Fp the rise of T r from one iteration to another may be slowed down to the extent which provides the convergence at the price of reducing its rate.…”

“…There are two comprehensive approaches that match this requirement. The first of them is the harmonic balance finite element method (HBFEM) [6,7], and the second one is based on the formulation of the time-dependent problem using constraints of time periodicity, known as the time-periodic finite element method (TPFEM) [8][9][10][11][12][13]. The HBFEM is recognized as being competitive with the transient timestepping model with respect to the execution time [7], although its accuracy may be limited when considering nonlinearity or high ordinals of time harmonics (switched currents).…”