Research And Application Of Domain Decomposition Method Based On Integral Equation

Electromagnetic simulation of modern electronic device system is of great significance for system performance evaluation and optimization design. However, most of these systems include complex geometric structures and complex materials, so such complicated system brings great challenge to the present numerical methods. Not only do they usually in need of large-scale electromagnetic field computations, but also they tend to have many very small features in the presence of electrically large structures. Such multi-scale electromagnetic problems tax heavily on numerical methods(finite element,finite difference, integral equation methods etc.) in terms of desired accuracy and stability of mathematical formulations. This dissertation presents approaches to analysis the multi-scale electromagnetic problems in the real-life applications using integral equation methods based on the domain decomposition method(IE-DDM), which is a kind of non-overlapping and non-conformal domain decomposition method. Compared with conventional integral methods, it is based on a divide-and-conquer philosophy. Instead of tackling a large and complex problem directly as a whole, the original problem is partitioned into smaller and easier to solve sub-domains. Some suitable boundary conditions called transmission conditions are prescribed at the interfaces between adjacent sub-domains to enforce the continuity of electromagnetic fields. On the one hand, suitable CEM solvers can be choose in various regions in the frame work of the DDM, which is endowed with the natural attributes of the parallelization. On the other hand, it extremely reduces the burden of preparation of geometry because DDM allows the mesh can be non-conformal. This is very important for the electromagnetic simulation of complex large problems. From the perspective of practical engineering application, the research work of this dissertation provides a flexible, numerical error controllable numerical tool for any perfect electric conductors, mulit-layer homogeneous dielectric cases, conductordielectric composers and thin dielectric coated targets.The dissertation started with the basic theorem of equivalence principle. Integral equation method is formulated by equivalence principle and boundary condition. Finally,the two kinds of basic integral operator are summed up. Namely, electrical field integral operator and magnetic field integral operator. In terms of the geometry modeling,mesh handling, basis functions and testing functions choosing and matrix equation solving these four aspects, we describe the proceeding of numerical solution of the integral equation. Then the physical quantity in the electromagnetic field and the corresponding function space is introduced and dual pairing principle is presented as the criteria for the selection of basis functions and testing functions.The non-conformal, non-overlapping integral equation domain decomposition method is formulated in detail. Furthermore, two kinds of iterative frame work based on innerouter iteration are studied and compared with each other in terms of iteration convergence. multi-level fast multipole algorithm(MLFMA) is also utilized to speed up the matrix-vector multiplier during the inner and outer iteration. At the same time, IE-DDM based on local-global MLFMA is proposed to address a special structure(body of translation) to achieve the fast and accurate solution effectively.Next, a reverse operation self-consistent evaluation(ROSE) is proposed to facilitate the implementation of mortar matrix in the non-conformal, non-overlapping domain decomposition method. Different from the direct transmission matrix filling method such as union mesh technique, this method recover the matrix entries reversely from known self-consistent samples. The sparsity pattern of the matrix is firstly analyzed, followed by a proper choice of a set of function samples that satisfies the transmission operation. A solvable system is constructed, and the matrix entries are then solved from these existing sample functions reversely in a self-consistent manner. This method totally bypasses the complicated direct evaluation in the traditional method. The error introduced is within the projection/interpolation operation, and the convergence with respect to the h-refinement agrees with the theoretical requirement in numerical analysis.Furthermore, this dissertation presents a hybrid approach combining non-conformal,non-overlapping domain decomposition method(DDM) and electric/magnetic current combined field integral equation(JMCFIE) to solve scattering problems of three dimensional multilayer dielectric objects and conductor-dielectric composition structure. The original complex structure can be decomposed into several closed sub-domains according to geometry feature and material distribution, and JMCFIE is formulated in each sub-domain. The major ingredients in the proposed approach are the use of the Robin type transmission conditions(TCs) and non-conformal, non-overlapping domain partitioning, which remarkably increases the flexibility of mesh discretization and enhances the efficiency of solution. MLFMA is also applied to realize fast computation of matrixvector product. When the computational region is conductor region, the solver can be turn to CFIE method easily. Here, four parts block diagonal precondition(4PBDP) can greatly reduce the number of inner iterations, so as to further improve the efficiency of the algorithm.Next, we investigate a new surface integral equation method based on discontinuous Garlerkin for the thin dielectric coating on conducting objects. It is based on a novel derivation of the weak formulation, in which the IBC is embedded. Both equivalent surface electric and magnetic currents are considered in the final formulations. It has a well-posed system matrix compared with the conventional methods. Also, the testing/trial functions can be expanded in the square-integrable space. The new surface integral equation is formulated by a proper dual paring to form a reaction integral, which is able to easily simulate scattering from objects with different IBCs, even from the perfect electric conductors(PEC) and the perfect magnetic conductors(PMC). Due to the discontinuous Galerkin scheme, it is possible to employ non-conformal surface discretization of the objects. It provides a new technical approach to solve complex electromagnetic scattering problems with thin dielectric coating.