A DOMAIN DECOMPOSITION METHOD FOR SOLVING ELECTRICALLY LARGE ELECTROMAGNETIC PROBLEMS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Kezhong Zhao, M.S. ***** The Ohio State University 2007 Dissertation Committee: Approved by Professor Jin-Fa Lee, Adviser Professor Fernando L. Teixeira ___________________________ Professor Ronald M. Reano Adviser Graduate Program in Electrical and Computer Engineering © Copyright by Kezhong Zhao 2007 ABSTRACT This dissertation presents a domain decomposition method as an effective and efficient preconditioner for frequency domain FEM solution of geometrically complex and electrically large electromagnetic problems. The method reduces memory requirements by decomposing the original problem domain into several non-overlapping and possibly repeatable sub-domains. At the heart of this research are the Robin-to-Robin map, the “cement” finite element coupling of non-conforming grids and the concept of duality paring. The Robin’s transmission condition is employed on interfaces between adjacent sub-domains to enforce continuity of electromagnetic fields and to ensure the sub-domain problems are well-posed. Through the introduction of cement variables, the meshes at the interface could be non-conformal which significantly relaxes the meshing procedures. By following the spirit of duality paring a symmetric system is obtained to better reflect physical nature of the problem. These concepts in conjunction with the so- called finite element tearing and interconnecting algorithm form the basic modules of the present domain decomposition method. To enhance the convergence of DDM solver, the Krylov solvers instead of classical stationary solvers are employed and studied. In order to account the radiation condition exactly thus eliminating spurious reflection, a boundary element formulation is hybridized with the present DD method, also through the aforementioned novel concepts. One of the special cases of present ii hybridization is the well known hybrid finite element and boundary element method. It will be shown that the proposed hybrid offers simultaneously: (1) symmetry, (2) modularity, (3) non-conformity between FEM and BEM domains, (4) free of internal resonance, and (5) natural and effective preconditioning scheme that guarantees spectral radius less or equal to one. Lastly this dissertation presents a DDM solution scheme for analyzing electromagnetic problems involving multiple separable scatterers. The method first decomposes the original problem into several disjoint sub-regions. In each sub-region, the domain decomposition method is further applied rendering geometrically complicated and electrically large sub-region problems tractable. The sub-regions communicate through the near-field Green’s function. To overcome the vast computational costs required in exchanging information between electrically large sub-regions, the adaptive cross approximation algorithm is adopted to expedite the process. iii Dedicated to My Family and the Loving Memory of My Late Wife, Xiu Ming Lin iv ACKNOWLEDGMENTS Fruitful harvest comes only with constant hard working. At this moment of celebrating my ending of long enduring student career and a beginning of new challenges, I wish to express my deepest appreciation to my advisor, Prof. Jin-Fa Lee, for introducing me into this fantastic subject, for sharing his passions, and for his guidance, continuous encouragement. I understand such appreciation can never be expressed in few simple words and will be with my heart throughout the remaining of my life. I would also like to express my deepest gratitude to the present and former members of the Computational Science Group at ElectroScience Laboratory, especially Dr. Marinos N. Vouvakis, Dr. Seung-Mo Seo, Seung-Cheol Lee, and Vineet Rawat, not only for their friendships, but for our stimulating discussions and their continuous encouragements throughout my graduate study. Thanks are extended to Prof. Jun Zou of Tsinghua University for his friendship and stimulating discussions during his time of visiting ElectroScience Laboratory. I wish to thank my examination committee members, Prof. Fernando L. Teixeira and Prof. Ronald M. Reano for their constructive comments, inputs and services throughout the Ph.D. examinations. In addition, I would like extend my thanks to the entire ElectroScience Laboratory family including fellow graduate students, staff, network and system administrators, researchers and professors for their helps and for v providing a wonderful research environment. Particularly, I would like to express my gratitude towards Prof. Prabhakhar H. Pathak, Prof. Denny Burnside, Dr. Teh-Hong Lee, Dr. Robert Burkholder for their kindness, teachings and collaborations. I am especially indebted to Prof. Denny Burnside and Dr. Teh-Hong Lee for our long-time collaborations, the permissions of using some of results, and providing some interesting real-life geometries along with their simulation results. I would also mention Mr. Kevin Reaver for continuous computer support and friendship. I would also express my gratitude to ANSOFT Corporation for the financial support of my last quarter’s tuition as well as their continuous support and interest on the research works over the years. Finally, I would like to express my most sincere gratitude to my parents and my wife, Xi Lin, for their understanding, patience, support and unconditional loves. vi VITA April 08, 1978 ................................................Born in Fuzhou, Fujian, China June 2001 B. S. Electrical Engineering, The Ohio State University, Columbus, OH. March 2003 ....................................................M.S. Electrical Engineering, The Ohio State University, Columbus, OH. March 2003 – present ....................................ElectroScience Laboratory, Electrical and Computer Engineering Department, The Ohio State University, Columbus, OH. PUBLICATIONS Research Publications K. Zhao, and J.-F. Lee, “A Single-Level IE-QR Algorithm to Model Large Microstrip Antenna Arrays”, IEEE Transactions on Antennas and Propagation, vol. 52, no. 10, pp. 2580-2585, Oct., 2004. M. N. Vouvakis, S.-C. Lee, K. Zhao and J.-F. Lee, “A Symmetric FEM-IE Formulation with a Single-Level IE-QR Algorithm for Solving Electromagnetic Radiation and Scattering Problems'”, IEEE Transactions on Antennas and Propagation, vol. 52, no. 11, pp. 3060-3070, Nov., 2004. S.-C. Lee, M. N. Vouvakis, K. Zhao and J.-F. Lee, “Analyzing Microwave Devices Using a Symmetric Coupling of Finite and Boundary Elements”, International Journal for Numerical Methods in Engineering, vol. 64, no. 4, pp. 528-546, Sept., 2005. K. Zhao, M. N. Vouvakis, and J.-F. Lee, “The Adaptive Cross-Approximation Algorithm for Accelerated Method of Moments Computations of EMC Problems”, IEEE Transactions on Electromagnetic Compatibility, vol. 47, no. 4, pp. 763-773, Nov., 2005. vii K. Zhao, M. N. Vouvakis, and J.-F. Lee, “Solving Electromagnetic Problems Using A Novel Symmetric FEM-BEM Approach”, IEEE Transactions on Magnetic, vol. 42, no. 4, pp.583-587, Apr., 2006. M. N. Vouvakis, K. Zhao, and J.-F. Lee, “FEM Analysis of Infinite Periodic Structures with Non-Matching Triangulations”, IEEE Transactions on Magnetic, vol. 42, no. 4, pp.691-695, Apr., 2006. M. N. Vouvakis, K. Zhao, S. M. Seo, and J.-F. Lee, “A Domain Decomposition Approach for Non-Conformal Couplings Between Finite and Boundary Elements for Unbounded Electromagnetic Problems in R3”, Journal of Computational Physics, vol. 225, no. 1, pp. 975-994, Jul. 2007. K. Zhao, V. Rawat, S.-C. Lee, and J.-F. Lee, “A Domain Decomposition Method with Non-Conformal Meshes for Finite Periodic and Semi-Periodic Structures”, IEEE Transactions on Antennas and Propagation, Sept. 2007. FIELDS OF STUDY Major Field: Electrical and Computer Engineering viii TABLE OF CONTENTS Page Abstract............................................................................................................................... ii Acknowledgments............................................................................................................... v Vita.................................................................................................................................... vii List of Tables .................................................................................................................... xii List of Figures..................................................................................................................xiii 1. Introduction......................................................................................................................1 1.1 Problem Statement.............................................................................................1 1.2 Background........................................................................................................2 1.3 Benefits of the DDM..........................................................................................3 1.4 Summary of Present DDM Algorithm...............................................................5 1.5 Notations............................................................................................................6 1.6 Outline................................................................................................................9 2. Non-Conforming Finite Element Domain Decomposition for Time-Harmonic Maxwell Equations............................................................................................................................11