Fast 3-D simulation of transient electromagnetic fields by model reduction in the frequency domain using Krylov subspace projection

Abstract

We present an efficient numerical method for the simulation of transient electromagnetic fields resulting from magnetic and electric dipole sources in three dimensions. The method we propose is based on the Fourier synthesis of frequency domain solutions at a sufficient number of discrete frequencies obtained using a finite element (FE) approximation of the damped vector wave equation obtained after Fourier transforming Maxwell's equations in time. We assume the solution to be required only at a few points in the computational domain, whose number is small relative to the number of FE degrees of freedom. The mapping which assigns to each frequency the FE approximation at these points of interest is a vector-valued rational function known as the transfer function. Its evaluation is approximated using Krylov subspace projection, a standard model reduction technique. Computationally, this requires the FE discretization at a small number of reference frequencies and the generation of a sufficiently large Krylov subspace associated with each reference frequency. Once a basis of this subspace is available, a sufficiently accurate rational approximation of the transfer function can be evaluated at the remaining frequencies at negligible cost. These partial frequency domain solutions are then synthesized to the time evolution at the points of interest using a fast Hankel transform. To test the algorithm, responses obtained by2-D and 3-D FE formulations have been calculated for a layered half-space and compared with results obtained analytically, for which we observed a maximum deviation of less than 2 per cent in the case of transient EM modelling. We complete our model studies with a number of comparisons with established numerical approaches. A first implementation of our new numerical algorithm already gives very good results using much less computational time compared with time stepping methods and comparable times and accuracy compared with the Spectral Lanczos Decomposition Method (SLDM).