Preconditioning iterative algorithms for electromagnetic scattering from large cavities

Abstract

The computation of electromagnetic problems has grown exponentially for three decades in engineering and applied mathematical fields. In this thesis, we investigate the electromagnetic scattering from a two-dimensional large open cavity embedded in an infinite ground plane, which is modelled by Maxwell or Helmholtz equations. By introducing transparent boundary conditions, the original problems defined on the infinite domain are reduced to bounded domain problems. Transparent boundary conditions are nonlocal. A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases, respectively. Finite difference discretizations of frequency domain models for problems with large wavenumbers always result in a large, sparse, symmetric, non-Hermitian, indefinite and ill-conditioned discrete system, for which direct methods are extremely expensive and classical iterative algorithms are slowly convergent or divergent. Because of its significant industrial and military applications, the large cavity problem has attracted much attention. In this thesis we focus on efficient numerical solution of electromagnetic scattering from cavities with inhomogeneous media and large wavenumbers. Firstly, we present several Toeplitz type approximations to the transparent boundary condition of hypersingularity for the TM case. These approximations give many advantages for developing fast algorithms to solve electromagnetic scattering from large cavities. Secondly, a fast algorithm of second-order approximation for the TM case is proposed to solve the cavity models with layered media. Moreover, two preconditioning techniques are developed for solving the cavity models with general media and non-rectangular empty cavity models. The first one is based on a simple physical cavity model with vertically layered media and the second one is based on a domain decomposition technique. Several Krylov subspace methods are employed to solve preconditioned systems. In addition, the sparse structure of the Krylov subspace and the symmetry of the preconditioners and the coefficient matrices for cavity models result in significant reduction of computational complexity. Finally, numerical results for several different cavity models with both real and complex media illustrate that these two proposed preconditioning algorithms are efficient, particularly the cavity models with large wavenumbers for which classical iterative algorithms cannot provide a convergence solution for non-preconditioned systems. Also numerical results indicate that BiCG requires less iterations than GMRES(m) and BiCGstab, particularly for large cavities.

Date of Award	2 Oct 2007
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Weiwei SUN (Supervisor)