Linear complexity direct finite element element solvers for general electromagnetic forward analysis and inverse design
Abstract
Driven by the design of advanced engineering systems, the complexity of computational electromagnetic methods need to be continuously reduced to meet the real-world challenges. Among existing methods, the best complexity of solving 3-D electromagnetic problems is O(N itNrhsN) for iterative methods and O( N2) for direct methods, where N is the matrix size, Nit is the number of iterations, and Nrhs is the number of right hand sides. Neither of the complexities has reached the optimal complexity of solving N parameters, which is O(N). In this thesis, we have developed a direct finite element solver of linear (O(N)) complexity for general 3-D electromagnetic analysis including both full-wave circuit extraction and the analysis of electrically large antennas. Both theoretical analysis and numerical experiments have demonstrated the solver's linear complexity in CPU time and memory consumption with prescribed accuracy satisfied. The proposed direct solver has successfully analyzed an industry product-level full package involving over 22.8588 million unknowns in approximately 16 hours on a single core running at 3 GHz. It has also rapidly solved large-scale antenna array of over 73 wavelengths with 3,600 antenna elements whose number of unknowns is over 10 million. The proposed direct solver has been compared with the finite element methods that utilize the most advanced direct sparse solvers and a widely used commercial iterative finite element solver. Clear advantages of the proposed solver in time and memory complexity as well as computational efficiency have been demonstrated. Modeling and simulation alone is not sufficient. It has to encompass an inverse design algorithm that can actually solve the design challenge. In view of such a need, we have extended the proposed linear-complexity direct solver to the fast synthesis of the physical layout of electromagnetic structures. In this method, we convert the layout synthesis to an equivalent inverse problem, and we avoid the need for an optimization procedure as well as the many forward runs of electromagnetic simulators involved in prevailing optimization methods. Numerical experiments have validated the proposed synthesis algorithm. The direct finite element solver developed in this work has been further extended to solve general sparse matrices resulting from a finite-element based analysis without mesh information. It is envisioned to benefit not only the area of electromagnetics but also other disciplines that require an efficient solution of large-scale sparse matrices.
Degree
Ph.D.
Advisors
Jiao, Purdue University.
Subject Area
Computer Engineering|Electrical engineering
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