Interpolating Multilevel Green’s Function with Radial Basis Function and Phase Compensation

As a kernel independent algorithm, multilevel Green’s function interpolation method (MLGFIM) [1] [2] has been developed for electromagnetic problems from low frequency to fullwave simulation. This method inherits the multilevel tree structure of multilevel fast multipole algorithm (MLFMA) and adopts the interpolation technique of pre-corrected fast Fourier transform (PFFT). Compare with other fast algorithms based on the method of moment (MoM), MLGFIM has advantages in flexibility for both multilayered and sparsely distributed multibody problems. Interpolation method as the key technique in MLGFIM affects the accuracy of this algorithm. In full-wave simulation, because of the oscillatory nature of the phase term of the Green’s function, it is not easy to approximate the Green’s function between two interaction cubes when the cube size is much larger than a wavelength. There are two kinds of interpolation schemes to be implemented in full-wave problem: one is to apply orthonormal radial basis function (RBF), and the other is to combine Lagrange interpolation with phase compensation (PC). The RBF of inverse multiquadric (IMQ) type in conjunction with staggered Tartan grid is used in [2]. However, due to the increasing condition number which follows with the increasing number of interpolation points, the cube length is restricted below two wavelengths. The comparison of different types of infinitely smooth RBFs [3] shows that the Gaussian (GA) RBF can obtain the best interpolation accuracy under the same condition [4]. From [4], although the interpolation of field points in a cube larger than two wavelengths is implemented, the number of interpolation points increases drastically to guarantee the error under the interpolation threshold. This will generate low interpolation efficiency when MLGFIM is applied. To remedy this problem, Lagrange interpolation with PC is adopted to make the phase terms of Green’s functions vary slowly with the distance between the source and field points [5], [6]. In this paper, we propose a method applying augmented RBF to interpolate Green’s function with PC. The results show that at the same interpolation threshold, interpolating Green’s function with PC requires fewer interpolation points than interpolating the original one. We also find that GA RBF can get the best convergence rate. But if the number of interpolation points is large, it is more appropriate to adopt RBF of compact support due to the fact that it will generate a sparse interpolation matrix.