Fast Computation Of The Sommerfeld Integrals In The Fast Multipole Method In Layered Media

In this paper,a high-precision numerical method for calculating a series of Sommerfeld integrals is proposed.The calculation of these Sommerfeld integrals is the key problem in the rapid calculation of Green function in layered media.The integrand of Green function usually has high oscillation,slow attenuation and singularity,and the direct application of traditional numerical integration methods is very inefficient,or even can not get the correct results.In this paper,by selecting an appropriate integral path in the complex plane,the integrand is ensured to have the fastest exponential descent along the new integral path while bypassing the singularity of the integrand.Then,the integral along the new path is calculated by using the numerical integral formula of unbounded region based on double-exponential(DE)transformation.Numerical experiments show that the proposed method achieves spectral precision and has very high efficiency.