Research On Electromagnetic Volume Integral Equation-Based Domain Decomposition Methods

Many areas of modern information technology involve solving complex electromagnetic problems, prompting the development of many efficient and practical electromagnetic algo-rithms. But, for complex dielectric structures of three dimensional electrically large, high medium parameter and/or high contrast ratio, the fast and effective analysis of their electro-magnetic scattering characteristics are still facing challenges. In order to cope with such chal-lenges, we in this dissertation research volume integral equation-based domain decomposition methods, and further explore the methods of reducing memory requirements, improving the computational efficiency and enhancing the convergence of inner-iteration in the domain de-composition method. The main contributions of this dissertation are as follows:1. Fast method of filling the system matrix of the volume integral equation-based method of moments is proposed. This method carries out comprehensive arrangement for all subdivision units to eliminate the redundant computation, reducing the matrix-filling time by about 80%.2. Fitting Green’s function FFT method based on the volume integral equation (VIE-FG-FFT) is proposed. The method accelerates the matrix-vector product by fitting the Green’s function onto nodes of a uniform Cartesian grid, whose computational complexity is O(N log N), and storage complexity is O(N), where N is the number of unknowns. Compared with previous FFT-based methods, the VIE-FG-FFT has the advantages of high accuracy and less pretreatment time. In particular, the procedure of fitting Green’s function has no relationship with medium parameters, so the application range of the VIE-FG-FFT can be extended to electrical anisotropic dielectric objects.3. Volume integral equation-based overlapping domain decomposition method (VIE-ODDM) with rigorous mathematical modeling procedure is proposed. This method transforms solving a global VIE-MoM model of an dielectric object of electrically large size solving many sub-domain models, and associates all sub-domain solutions with each other by us-ing an "outer iteration scheme", greatly reduce the memory requirement, and hence, it can solve the electromagnetic scattering from the electrically large inhomogeneous dielectric objects whose usual fast algorithm models are too large for the user’s computer to accom-modate. In particular, the convergence of the outer-iteration scheme of the VIE-ODDM is investigated theoretically and numerically, and confirmed to be very good. Finally, the application range of the VIE-ODDM is extended to electrical anisotropic dielectric objects.4. Hybrid method of the volume integral equation-based overlapping domain decomposition method and fitting Green’s function fast Fourier transform method (VIE-ODDM-FG-FFT) is proposed. This method not only maintains the advantages of the VIE-ODDM, but also further reduces the memory requirements and greatly improves the computa-tional efficiency. Compared with the scheme using the MLFMA as the accelerator, this method has no the problem of sub-wavelength breakdown. Besides, due to introducing the nested uniform Cartesian grid scheme, the computational efficiency of this method won’t be significantly affected by uneven medium parameter distributions, even high-contrast structures.5. Volume integral equation-based non-overlapping domain decomposition method (VIE-NDDM) and its combination with the FG-FFT method (VIE-NDDM-FG-FFT) are pro-posed. Unlike the VIE-ODDM, the VIE-NDDM uses the explicit boundary condition to build information coupling between adjacent subdomains, avoiding constructing buffer regions, and hence, reducing pretreatment time. Similar to the VIE-ODDM-FG-FFT, the introduction of the FG-FFT further reduces the memory requirement and greatly improves the computational efficiency.