AbstractsComputer Science

This dissertation presents approaches to solve the multi-scale and wideband problems using surface integral equation methods based on the skeletonalization technique, which in essence identifies the numerically independent elements from a larger set of unknowns. In the low frequency or multi-scale scenario, overly dense mesh is generated in a global or local scale. The method is extended to composite material through the integral equation discontinuous Galerkkin method via enhanced enforcement of transmission conditions. Conventional multi-level fast multipole method(MLFMM) faces low frequency breakdown since a large number of basis functions are concentrated within the leaf level groups, whose size is typically larger than ¿/4. The computational complexity rapidly approaches that of conventional MoM, which is O ( N ²) for both CPU time and memory consumption for iterative solvers. In this dissertation a hierarchical multi-level fast multipole method (H-MLFMM) is proposed to accelerate the matrix-vector multiplication for low frequency and multi-scale problems. Two different types of basis functions are proposed to address these two different natures of physics corresponding to the electrical size of the elements. Moreover, the proposed H-MLFMM unifies the procedures to account for the couplings using these two distinct types of basis functions. O ( N ) complexity is observed for both memory and CPU time from a set of numerical examples with fixed mesh sizes. Numerical results are included to demonstrate that H-MLFMM is error controllable and robust for a wide range of applications. On the other hand, condition number of the system matrix deteriorates due to the overly dense mesh. This would greatly affect the convergence of iterative solvers, if convergence can ever be attained. Direct solver This thesis proposes an algorithm exploits the smoothness of the far field and computes a low rank decomposition of the off-diagonal coupling blocks of the matrices through a set of skeletonalization processes. Moreover, an artificial surface (the Huygens' surface) is introduced for each clustering group to efficiently account for the couplings between well-separated groups. Furthermore, a recursive multi-level version of the algorithm is developed subsequently. Through numerical examples, we found that the proposed multi-level direct solver can scale as good as O ( N ^1.3) in memory consumption and O ( N ^1.8) in CPU time, for moderate-sized EM problems as the electrical size grows. An novel IEDG method with enhanced enforcement of transmission conditions is proposed based on the IEDG algorithm scheme, this makes it possible to solve surface integral equation without being confined to conformal mesh and basis functions with inter-element continuity. Basis functions with different definitions and polynomial orders can be mixed flexibly to form a robust surface integral equation solver for multi-scale structures. IEDG algorithm allows local mesh…