Fast Reduction Algorithms in the Frequency-Domain Layered Finite Element Method for the Electromagnetic Analysis of Large-Scale High-Frequency Integrated Circuits

In this paper, fast algorithms are proposed for an efficient reduction of a 3-D layered system matrix to a 2-D layered one in the framework of the frequency-domain layered finite element method. These algorithms include: 1) an effective preconditioner
P
that can converge the iterative solution of the volume-unknown-based matrix equation in a few iterations; 2) a fast direct computation of
P

^-1
in linear complexity in both CPU run time and memory consumption; and 3) a fast evaluation of
P

^-1

b
in linear complexity, with
b
being an arbitrary vector. With these fast algorithms, the volume-unknown-based matrix equation is solved in linear complexity with a small constant in front of the number of unknowns, and hence significantly reducing the complexity of the 3-D to 2-D reduction. The algorithms are rigorous without making any approximation. They apply to any arbitrarily-shaped multilayer structure. Numerical and experimental results are shown to demonstrate the accuracy and efficiency of the proposed algorithms.