Abstract
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 <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</b> that can converge the iterative solution of the volume-unknown-based matrix equation in a few iterations; 2) a fast direct computation of <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</b> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> in linear complexity in both CPU run time and memory consumption; and 3) a fast evaluation of <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P</b> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> in linear complexity, with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</i> 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.
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