A Two-Step Method Coupling Eddy Currents and Magneto-Statics

TL;DR

The paper tackles the challenge of including eddy-current effects only in a subdomain of a larger domain, forming a heterogeneous coupling with magneto-static regions. It introduces a two-domain two-step method that decouples the electric scalar potential $\\varphi$ and the magnetic vector potential $\\mathbf{A}$ using a DC-conduction gauge and electric circuit element boundary conditions, solved sequentially to yield $\\varphi$ followed by $\\mathbf{A}$. Numerical validation on a conductive cylinder in air demonstrates expected convergence and shows how to reconstruct the port voltages, either directly from potentials or from the total power $P_{total}$ when eddy currents are confined. The framework reduces computational complexity in scenarios with moving components by avoiding full eddy-current time-stepping in the remeshed region while maintaining accurate Lorentz-force and magnetic-field predictions for switching devices.

Abstract

We present the mathematical theory and its numerical validation of a method tailored to include eddy-current effects only in a part of the domain. This results in a heterogeneous problem combining an eddy-current model in a subset of the computational domain with a magneto-static model in the remainder of the domain. We adopt a two-domain two-step approach in which the primary variables of the problem are the electric scalar potential and the magnetic vector potential. We show numerical results that validate the formulation.

Figures (6)

• Figure 1: Test case 1: (a) convergence in $L^2$-norm and (b) convergence in $H(\hbox{curl})$-seminorm.
• Figure 2: Test case 2 and 3 (prisms): Three portion inner cylinder. The blue and the red portions refers to the iron part, whereas the gray portion refers to the copper part.
• Figure 3: Test case 2: Evolution in time of $L^2$-norm of difference in current density between numeric solution on three-portion cylinder and analytic solution on infinite cylinder: (a) in first (or third) portion of inner cylinder (iron) and (b) in second portion of inner cylinder (copper).
• Figure 4: Test case 3: Evolution in time of $L^2$-norm of difference in current density between numeric solution on three-portion cylinder and analytic solution on infinite cylinder: (a) in first (or third) portion of inner cylinder (iron) and (b) in second portion of inner cylinder (copper).
• Figure 5: Test cases 2 and 3: cylinder $\mathcal{C}_5$ in refinement $\mathcal{N}_{\mathrm{ref}3}$: comparison of the normalized voltage computed using (\ref{['Busetto:eq:total_voltage']}) for Test cases 2 and 3, and the voltage reconstructed from the power for Test case 2.