Discretized Boundary Equation Method And Its Applications

A unified approach, named discretized boundary equation (DBE) method, is introduced in this dissertation. The DBE's can be used either on the object surface to obtain the solution directly or on the truncation boundary of a finite difference (FD) or finite element (FE) mesh as termination conditions.A new on-surface formulation, namely OS-DBE, is presented as an application of the two-component version of the DBE method to generate the solutions of two-dimensional (2D) scattering by perfect electric conducting (PEC) cylinders and three-dimensional (3D) electrostatics problems. Different from other numerical methods, the OS-DBE allows independent determination of the field at any given point on the object surface with a matrix of smaller order than in the method of moment (MoM). Hence, the method is very suitable for parallel computing and the numerical efficiency for the calculation of the source distribution can be further improved through the asymptotic waveform evaluation (AWE). The OS-DBE can be combined with the fast multipole method (FMM) or the multilevel FMM (MLFMM) to accelerate the solution of the matrix equations. When the basis functions and field points cover the whole object surface, the solution determined by the OS-DBE method is more accurate than that obtained directly by the MoM solution of the electric field integral equation or the magnetic field integral equation with the same mesh size. So the present OS-DBE also provides a way to improve the accuracy or efficiency of the MoM. In addition, the present OS-DBE can avoid the resonance phenomenon and provides an alternative way to eliminate resonance solutions in the MoM.In this dissertation, the OS-DBE method is combined with the AWE technique and applied to 2D scattering and 3D electrostatic problems to sweep the source distribution in the spatial domain, so as to further improve the numerical efficiency.The DBE is also used as mesh termination condition for the FD or the FE method. One kind of termination condition for 2D problems is derived based on the one-component formulation of the DBE method. The DBE in this case has the following relation to the measured equation of invariance (MEI): The DBE obtained with the minimum norm least squares (MNLS) method is equivalent to the MEI generated by the minimum norm (MN) technique, while the DBE produced by the MN method corresponds to the MEI using MNLS technique. In this dissertation, the DBE-MNLS technique is investigated thoroughly and the related numerical techniques are improved. Although this technique inevitably sacrifices the sparsity of the overall matrix, it is well compensated by the significantly improved performance in the matrix solution and the overall efficiency is still much better. 2D scattering and 3D electrostatics problems are solved by the DBE-MNLS. The other kind of mesh termination condition for 2D problems is derived based on the two-component formulation of the DBE method. This new termination condition is used to obtain the solutions of 2D scattering problems by PEC cylinders. This efficient method can generate a sparse matrix with good condition number while the sparsity should be further improved.When the DBE is imposed on the truncation boundary in the FD method for 3D scattering by PEC objects, a linear tensor relation for the vector field in the DBE can be derived instead of a simple uncoupled linear relation for each component of the vector field as in 2D cases. The DBE-MNLS technique is still used to generate numerical results.