Integral equation solution for the transient electromagnetic response of a three‐dimensional body in a conductive half‐space

Abstract

The time‐domain integral equation for the three‐dimensional vector electric field is formulated as a convolution of the scattering current with the tensor Green’s function. The convolution integral is divided into a sum of integrals over successive time steps, so that a numerical scheme can be formulated with a time stepping approximation of the convolution of past values of the solution with the system impulse response. This, together with spatial discretization, leads to a matrix equation in which previous solution vectors are multiplied by a series of matrices and fed back into the system by adding to the primary field source vector. The spatial discretization, based on a modification of the usual pulse basis formulation in the frequency domain, includes an additional subset of divergence‐free basis functions generated by integrating the Green’s function around concentric closed rectangular paths. The inductive response of the body is more accurately modeled with these additional basis functions, and a meaningful solution can be obtained for a body in free space. The resulting algorithm produces good results even for large conductivity contrasts. Internal checks, including convergence with respect to spatial and temporal discretization, and reciprocity, demonstrate self‐consistency of the numerical scheme. Independent checks include (a) comparison with results computed for a prism in free space, (b) comparison with results computed for a thin plate, (c) comparison of our conductive half‐space algorithm with an asymptotic solution for a sphere, and (d) comparison with results from inverse Fourier transformation of values computed using a frequency‐domain integral equation algorithm. Qualitative features of the results show that the relative importance of current channeling and confined eddy currents induced in the body depends upon both conductivity contrast and geometry. If the free‐space time constant is less than the time window during which currents in the host have not yet propagated well beyond the body, current channeling dominates the response. In such cases, simple superposition of free‐space results and the background is a poor approximation. In cases where the host currents diffuse beyond the body in a time less than the free‐space time constant of the body, the total response is approximately the sum of the free‐space and background (half‐space) responses.