Saving Memory With Vector Addition Theorem

Abstract

Abstract— In the low-frequency fast multipole algorithm (LF-FMA), instead of the traditional factorization of the scalar Green’s function using scalar addition theorem, we adopt the vector addition theorem for the factorization of the dyadic Green’s function to realize the memory savings. Recently, the factorization and the diagonalization of the dyadic Green’s function is possible based entirely on the use of vector wave functions and vector addition theorem [1, 2]. In this work, we shall validate this factorization and diagonalization. This factorization can be used to develop a vector mixed-form fast multipole algorithm for the dyadic Green’s function [3]. By using the new integral operator factorization with vector addition theorem and loop-tree basis, every element of the matrix resulting from applying the method of moments (MoM) to the electric field integral equation (EFIE) can be expressed by the radiation patterns, translator and receiving patterns. The storage of the radiation patterns and receiving patterns can be reduced by 25 percent compared with the one when the factorization of the scalar Green’s function is used. In addition, since the elements of the new translator can be calculated on the fly from the elements of the translator given by the factorization of the scalar Green’s function using scalar addition theorem, we need only to store the latter. Therefore the whole memory usage is reduced and the memory saving is important for the large-scale computation with finite computer resource. REFERENCES 1. Chew, W. C., “Vector addition theorem and its diagonalization,” Comm. Comput. Phys., Vol. 3, No. 2, 330–341, Feb. 2008. 2. He, B. and W. C. Chew, “Diagonalizations of vector and tensor addition theorems,” Comm. Comput. Phys., Vol. 4, No. 4, 797–819, 2008. 3. Jiang, L. J. and W. C. Chew, “A mixed-form fast multipole algorithm,” IEEE Trans. Antennas Propag., Vol. 53, No. 12, 4145–4156, Dec. 2005.