In this paper, a fast multipole expansion technique (FMT) is applied to the desingularized boundary element method (DBEM) to solve three -dimensional water wave problems. Besides avoiding the existence of singularity integration, computing time and memory storage are decreased to about O(NlgN) and O(N) order respectively, where N is the unknown number. Numerical experiments are carried out to validate the efficiency and accuracy of the present method. The present model is used to calculate linear and fully nonlinear wave problems: surge radiation of a submerged sphere and motion of a source-sink pair under the free surface. And good results are obtained by comparison with the known ones.
As a popular and convenient method, the boundary element method has been widely used to solve various water wave problems in both frequency domain (e.g. Nossen et al, 1991; Teng and Eatock Taylor, 1995b) and time domain (e.g. Kim and Kim, 1997; Isaacson and Cheung, 1992). However, because of its inherent shortcomings, singularity and O(N2) order calculation time and memory storage, the application of the conventional BEM has been limited to a great extent. In order to overcome above bottlenecks, some smart ideas have been invented to overcome or avoid singularity in surface integration, such as mapping technique (Zang et al,1995), rigid mode method (Brebbia, 1978) and desingularized boundary element method (Beck, et al, 1993); Meanwhile, there have been many successful efforts to reduce the numerical operation and storage requirements and introduce fast algorithm, such as the use of wavelet basis (Alpert et al, 1993), the adoption of FFT (Tsinopoulos et al, 1999) and the advent of FMT (Rokhlin,1985) etc, among which the FMT is a particularly promising method. FMT can reduce the computational cost and the memory requirements to about O(N lgN) and O(N) order respectively.
