Regularization and fast solution of large subwavelength problems with imperfectly conducting materials

We describe an approach to low-frequency regularization of surface integral equations. We consider integral equations modeling general electrically and magnetically conducting sheets characterized by first-order boundary conditions, and parameterized in terms of electric, magnetic, and "cross" resistivities. Special cases of such boundary conditions include perfect conductors, resistive and impedance sheets, and thin penetrable sheets. Our approach constitutes an extension/generalization of the well-established regularization method for problems involving perfect conductors, based on rescaling of solution components in the solenoidal (loop) and remainder (e.g., tree;) subspaces. We demonstrate, on several examples of practical interest, that the proposed method leads to a significant improvement of the integral equation stability for problems with thin surface material sheets, through the use of a rescaling procedure different from that for perfect conductors, and controlled by the frequency dependence of the material parameters. The proposed method significantly improves conditioning of the impedance matrix, and results in correspondingly accelerated convergence ad iterative solutions.We also describe two oilier aspects of our approach: (a) Construction of an efficient algorithm allowing identification of loop and tree solution subspaces, applicable to locally two-dimensional surfaces of general topology (with possible boundaries and handles). The algorithm complexity is approximately O(N), where N is the number of faces of the mesh. (b) Interfacing of the low-frequency regularization with fast FFT-based (ATM) solver, which allows us to treat large problems (of sub-wavelength type) and provides a smooth transition to arbitrarily low frequencies. We show examples of application of the approach to solution to low frequency magnetic shielding problems involving geometrically and topologically complex configurations of high permeability materials.