A Hybrid a–φ Cell Method for Solving Eddy–Current Problems in 3–D Multiply–Connected Domains

A hybrid $a - \varphi $ Cell Method formulation for solving eddy–current problems in 3–D multiply–connected regions is presented. By using the magnetic scalar potential the number of degrees of freedom in the exterior domain with respect to the $A,V - A$ formulation, typically implemented in commercial software for electromagnetic design, can be almost halved. On the other hand, the use of the magnetic vector potential in the interior domain improves the flexibility with respect to $T - \Omega $ formulation, since both conductive and magnetic parts can be easily modeled. By using a Cell Method variant, based on an augmented dual grid for discretization, electric and magnetic variables can be consistently coupled at the interface between interior and exterior domain. Global basis functions needed for representing the magnetic field in the insulating region are obtained by using for the first time iterative solvers relying on auxiliary space preconditioner and aggregation–based algebraic multigrid, with linear optimal complexity. These represent highly–efficient alternatives to traditional computational topology algorithms based on the concept of thick cut. As a result, an indefinite symmetric matrix system, amenable to fast iterative solution, is obtained. Numerical tests show high accuracy and fast convergence of the $a - \varphi $ method on test cases with complex topology. Computational cost for both matrix assembly and linear system solution is limited even for large problems. Comparisons show that the $a - \varphi $ method provides better performance than existing methods such as $A,V - A$ and $h - \varphi $ .