A cylindrical H(curl)-conforming finite element: Two-domain method for wire modeling with segmentation

Efficient simulations of multiscale problems using the finite element method require specific adjustments. An example is the eddy current problem of a winding with only a few turns and a non-negligible skin effect and proximity effect. Brute force finite element modeling of the thin wire and a small penetration depth may become rather expensive or practically impossible. In this case, homogenization of the winding is also not a feasible method. There is an obvious lack in modeling and simulation of such wire problems. To this end a novel two-domain method is developed to solve the problem at hand. The few turns of the winding are one domain, and the rest of the problem is the other. To allow for a convenient modeling of the winding the wire is assumed to be divided into cylindrical segments. Therefore, a hierarchical H ( curl ) -conforming cylindrical finite element corresponding to a wire segment was developed to avoid very fine meshes. The construction of the hierarchical H ( curl ) -conforming cylindrical finite element basis is described in detail. Outside the wire, the solution for the rest of the problem is approximated with sufficient accuracy using the standard finite element method on a relatively coarse mesh. The magnetic vector potential A is used to solve the eddy current problem. A domain decomposition method using a Lagrange multiplier is exploited to enforce weak continuity of the solution across the interface between air and wire. Numerical examples with numerous investigations demonstrate the effectiveness and accuracy of the proposed two-domain method in comparison to reference solutions based exclusively on the standard finite element method using h - and p -refinement.