Boundary Element Solution of Electromagnetic Fields for Non-Perfect Conductors at Low Frequencies and Thin Skin Depths

A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximation is developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the quasistatic approximation. A two-term asymptotic expansion with respect to a small parameter characterizing the skin depth is derived for the magnetic and electric fields outside and inside the scatterer, which can be extended to higher order terms if needed. The introduction of a special surface operator (the inverse surface gradient) allows the reduction of the computational complexity of the solution. A method to compute this operator is developed. The obtained formulation operates only with scalar quantities and requires the computation of surface integral operators that are customary in boundary element (method of moments) solutions to the Laplace equation. The formulation can be accelerated using the fast multipole method. The resulting method is much faster than solving the vector Maxwell equations. The obtained solutions are compared with the Mie solution for scattering from a sphere, and the error of the solution is studied. Computations for much more complex shapes of different topologies, including for magnetic and electric field cages used in testing, are also performed and discussed.