Modular Fast Direct Solvers for Electromagnetic Modeling

Significant recent progress has been made in developing fast, direct solution methods for electromagnetic applications. For example, it has recently been demonstrated that local-global solution (LOGOS) modes provide an effective framework for developing efficient direct solvers for both low and high frequency electromagnetic problems with nearly optimal asymptotic computational complexities. The purpose of the presently proposed effort is to further exploit and build on the novel features of the LOGOS-based direct solution framework in order to develop modular and reduced-order solvers for large-scale, three-dimensional electromagnetic modeling applications in the frequency domain. The LOGOS direct solution framework is based on the expansion of the underlying system matrix in a complete basis of local solutions that satisfy global constraints. An advantage of this approach is that both the system matrix and its inverse are sparse in the LOGOS basis. This feature has been exploited to develop fast, direct solution methods for large electromagnetic modeling problems. However, in addition to providing efficient direct solvers, the LOGOS framework also leads naturally to the development of error-controllable algorithms for full-wave modular and reduced-order modeling. In particular, the LOGOS expansion provides a welldefined mechanism for the controllable identification of all solution modes on a large simulation domain that are also independent of geometric modifications made within a specific design volume. Within the proposed effort, this capacity will be used to develop novel modular solution algorithms for rapid electromagnetic analysis and design of systems and devices located on large platforms. It is expected that the resulting solution algorithms will allow one to analyze the impact of local design modifications on the performance of the larger system much more rapidly than would be possible if the system were to be re-factored for each design modification. In addition to this, due to the unique properties of the underlying LOGOS expansion, it will be possible to perform the initial factorization without knowing a priori where the design regions will subsequently be located. The latter capability is not provided by existing simulation technologies and is expected to be of significant value in the design and analysis of extremely large platforms by a team of engineers. The modular solution methods discussed above can be further extended to develop errorcontrollable full-wave reduced-order models for general engineering design problems. In many such scenarios, an engineer is interested in modeling the response of a system to specific modifications (e.g., changes in geometry, materials, etc.). Using existing simulation technologies, it is necessary to re-compute all N of the underlying degrees of freedom (e.g., the fields on the entire domain) in order to extract the quantities of interest (e.g., the maximum field strength in a particular region). Within the presently proposed effort, we will exploit the natural locality of the LOGOS expansion in conjunction with the modular solver mentioned above to develop and demonstrate a general procedure for extracting error-controlled reduced-ordered models for specific, user-specified input-output relationships. The resulting reduced-order models will allow an engineer to perform effectively in situ design cycles for sub-systems located on extremely large platforms in fewer than O(N) operations. This possibility is not provided by existing full-wave simulation technologies. 3