A low-frequency fast multipole boundary element method for acoustic problems in a subsonic uniform flow

The fast multipole method (FMM) based on the plane wave expansion is known to suffer from numerical instability in the low-frequency regime. This paper presents a low-frequency fast multipole boundary element method (LF-FMBEM) for acoustic problems in a subsonic uniform flow. First, a hybrid convected boundary integral formula based on the Burton-Miller method is derived to overcome the non-uniqueness difficulty at fictitious eigenfrequencies. The explicit evaluation of hypersingular integrals in the convected boundary integral formulae is also introduced. Then, the formulae of FMM based on the series expansion for convected BEM are derived to improve the calculation efficiency. The recursive calculation method is derived in the expansion of the derivative of the integrands. Besides, the rotation-coaxial translation-rotation back (RCR) technique is employed to accelerate the multipole translation. The numerical implementation process of the developed algorithm is presented in detail. Several numerical experiments are performed to validate the computational efficiency and accuracy of the developed LF-FMBEM. Results show that the proposed algorithm can achieve large-scale computation of one million degrees of freedom (DOF) on a personal computer, and the computational accuracy is still high when the Mach number reaches 0.95. The non-uniqueness problem for convected acoustics problems is also effectively overcome.