Simulation of electromagnetic induction scattering from targets with negligible to moderate, penetration by primary fields

The problem of numerical modeling of electromagnetic induction (EMI) responses by metallic objects is complicated by the fact that transmitted fields may penetrate the target, but will often only do so slightly. The effect cannot be ignored, yet it is often grossly impractical to discretize the entire surface or volume of a target in space increments only on the order of a fraction of the skin depth. To deal with this problem, we retain a simple integral equation formulation in scalar potential for the region outside the target, where magnetic fields are quasi-static and irrotational. Within the target we apply only the divergence relation, del.H = 0. When the skin depth is small relative to the radius of curvature of the target (e.g., <0.1), we use the thin skin depth approximation (TSA), partial derivativeH(n)/partial derivativen as similar toikH(n), just inside the target's surface, where k is the electromagnetic wavenumber inside the metal and n is the normal direction on the surface and pointing inside of metallic object. Examination of analytical solutions for the sphere suggests the parameter range in which this approximation might perform well and suggests ways of improving accuracy over an extended range. The fundamental TSA formulation appears to be relatively robust. Analysis indicates that it is insensitive to variation over the target's surface of primary field orientation relative to that surface, and that it is only dependent on the target's magnetic permeability through induction number. Implementing the TSA numerically, within the above divergence relation, allows us to express all quantities in terms of tangential magnetic field components and their tangential derivatives over the target surface. In principle, this closes the system completely in terms of the exterior scalar potential. Broad-band numerical simulations based on the TSA compare favorably with analytical and other numerical solutions. Test cases involving negligible skin depth show some fundamental induction scattering sensitivities, or lack thereof, for spheroidal and ellipsoidal target geometries and deformations of them. Tests are also performed for prolate spheroidal targets, over a wide range of magnetic permeabilities, for frequencies spanning the orders of magnitude characteristic of current broad-band EMI measurement systems.