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Addition Theorem

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Weng Cho Chew – 1st expert on this subject based on the ideXlab platform

  • A Low Frequency Vector Fast Multipole Algorithm with Vector Addition Theorem
    Communications in Computational Physics, 2020
    Co-Authors: Weng Cho Chew

    Abstract:

    In the low-frequency fast multipole algorithm (LF-FMA) [19, 20], scalar Addition Theorem has been used to factorize the scalar Green’s function. Instead of this traditional factorization of the scalar Green’s function by using scalar Addition Theorem, we adopt the vector Addition Theorem for the factorization of the dyadic Green’s function to realize memory savings. We are to validate this factorization and use it to develop a low-frequency vector fast multipole algorithm (LF-VFMA) for lowfrequency problems. In the calculation of non-near neighbor interactions, the storage of translators in the method is larger than that in the LF-FMA with scalar Addition Theorem. Fortunately it is independent of the number of unknowns. Meanwhile, the storage of radiation and receiving patterns is linearly dependent on the number of unknowns. Therefore it is worthwhile for large scale problems to reduce the storage of this part. In thismethod, the storage of radiation and receiving patterns can be reduced by 25 percent compared with the LF-FMA. © 2010 Global-Science Press.link_to_subscribed_fulltex

  • efficient ways to compute the vector Addition Theorem
    Journal of Electromagnetic Waves and Applications, 1993
    Co-Authors: Weng Cho Chew, Y M Wang

    Abstract:

    Two efficient ways of calculating the vector Addition Theorem are presented. One is obtained by relating the coefficients of the vector Addition Theorem to that of the scalar Addition Theorem for which an efficient recurrence relation exists. The second way is to derive recurrence relations directly for the coefficients of the vector Addition Theorem. These new ways of calculating the coefficients are of reduced computational complexity. Hence, when the number of coefficients required is large, the present methods are many times faster than the traditional method using Gaunt coefficients and Wigner 3j symbols.

  • recurrence relations for three dimensional scalar Addition Theorem
    Journal of Electromagnetic Waves and Applications, 1992
    Co-Authors: Weng Cho Chew

    Abstract:

    Recurrence relations for the elements of a translation matrix in the scalar Addition Theorem in three-dimensions using spherical harmonics are derived. These recurrence relations are more efficient to evaluate compared to the use of Gaunt coefficients evaluated with Wigner 3j symbols or with recurrence relations. The efficient evaluation of the Addition Theorem is important in a number of wave scattering calculations including fast recursive algorithms.

W. C. Chew – 2nd expert on this subject based on the ideXlab platform

  • The tensor Addition Theorem: From the viewpoint of group theory
    2008 IEEE Antennas and Propagation Society International Symposium, 2008
    Co-Authors: B. He, W. C. Chew

    Abstract:

    In this paper, the tensor Addition Theorem is discussed. Some familiarity with the theory of Lie group is also presented.

  • Diagonalization of the vector Addition Theorem
    2007 IEEE Antennas and Propagation Society International Symposium, 2007
    Co-Authors: W. C. Chew

    Abstract:

    The development of fast algorithms for integral equation solvers opens up new realms for the applications of integral equation solvers [V. Rokhlin, 1990; C.C. Lu and W.C. Chew, 1994; J.M. Song and W.C. Chew, 1995; W.C. Chew et al., 2001]. One of these is their use in the arena of circuits, micro-circuits and nanotechnologies. Often time, the use of fast solvers in this arena calls for the combined use of fast algorithms where both wave physics and circuit physics are captured well by the solvers [L.J. Jiang and W.C. Chew, 2005].

  • An efficient way to compute the vector Addition Theorem [EM scattering]
    Proceedings of IEEE Antennas and Propagation Society International Symposium, 1993
    Co-Authors: Y M Wang, W. C. Chew

    Abstract:

    In this method, the coefficients of the vector Addition theory are obtained by relating them to that of the scalar Addition Theorem for which an efficient recurrence relation has been derived by Chew (1992). This new scheme of calculating the coefficients is of reduced computational complexity. Hence, when the number of coefficients required is large, the present methods are many times faster than the traditional method using Gaunt coefficients and Wigner 3j symbols.

J. J. Tsai – 3rd expert on this subject based on the ideXlab platform

  • Equivalence Between The Trefftz Method AndThe Method Of Fundamental Solutions ForGreen’s Function Of Concentric Spheres UsingThe Addition Theorem And Image Concept
    Mesh Reduction Methods, 2009
    Co-Authors: J. T. Chen, H. C. Shieh, J. J. Tsai

    Abstract:

    Following the success of the mathematical equivalence between the Trefftz method and the method of fundamental solutions for the annular Green’s function, we extend to solve the Green’s function of 3-D problems in this paper. The Green’s function of the concentric sphere is first derived by using the image method which can be seen as a special case of method of fundamental solutions. Fixed-fixed boundary conditions are considered. Also, the Trefftz method is employed to derive the analytical solution by using the T-complete sets. By employing the Addition Theorem, both solutions are found to be mathematically equivalent when the number of Trefftz bases and the number of image points are both infinite. In the successive image process, the final two images freeze at the origin and infinity, where their singularity strengths can be analytically and numerically determined in a consistent manner. The agreement among the three results, including two analytical solutions by using the Trefftz method and the image method, and one numerical solution by using the conventional MFS is observed.

  • equivalence between the trefftz method andthe method of fundamental solutions forgreen s function of concentric spheres usingthe Addition Theorem and image concept
    WIT Transactions on Modelling and Simulation, 2009
    Co-Authors: J. T. Chen, H. C. Shieh, J. J. Tsai

    Abstract:

    Following the success of the mathematical equivalence between the Trefftz method and the method of fundamental solutions for the annular Green’s function, we extend to solve the Green’s function of 3-D problems in this paper. The Green’s function of the concentric sphere is first derived by using the image method which can be seen as a special case of method of fundamental solutions. Fixed-fixed boundary conditions are considered. Also, the Trefftz method is employed to derive the analytical solution by using the T-complete sets. By employing the Addition Theorem, both solutions are found to be mathematically equivalent when the number of Trefftz bases and the number of image points are both infinite. In the successive image process, the final two images freeze at the origin and infinity, where their singularity strengths can be analytically and numerically determined in a consistent manner. The agreement among the three results, including two analytical solutions by using the Trefftz method and the image method, and one numerical solution by using the conventional MFS is observed.