The direct BIE formulation and its numerical solutions using the BEM for 2D elasticity problems were developed by Rizzo in the early 1960s and published in Ref. [4] in 1967. Following this early work, extensive research efforts were made for the development of the BIE and BEM for solving various elasticity problems (see, e.g., Refs. [24–28]). The advantages of the BEM for solving elasticity problems are the accuracy in modeling stress concentration or fracture mechanics problems and the ease in modeling complicated elastic domains such as various composite materials.

The FMM was applied to solving elasticity problems for more than a decade. For 2D elasticity problems, Greengard et al. [68, 69] developed a fast multipole formulation for solving the biharmonic equations using potential functions. Peirce and Napier [36] developed a spectral multipole approach that shares some common features with the FMMs. Richardson et al. [70] proposed a similar spectral method using both 2D conventional and traction BIEs in the regularized form. Fukui [71] and Fukui et al. [72] studied both the conventional BIE for 2D stress analysis and the HBIE for large-scale crack problems. In his work, he first applied the complex variable representation of the kernels and then used the multipole expansions in complex variables as originally used for 2D potential problems [35, 62]. Liu [73, 74] further improved Fukui's approach and proposed a new set of moments for 2D elasticity CBIEs, which yields a very compact and efficient formulation with all the translations being symmetrical regarding the two sets of moments.

This book is an introduction to the fast multipole boundary element method (BEM), which has emerged in recent years as a powerful and practical numerical tool for solving large-scale engineering problems based on the boundary integral equation (BIE) formulations. The book integrates the classical results in BIE formulations, the conventional BEM approaches applied in solving these BIEs, and the recent fast multipole BEM approaches for solving large-scale BEM models. The topics covered in this book include potential, elasticity, Stokes flow, and acoustic wave problems in both two-dimensional (2D) and three-dimensional (3D) domains.

The book can be used as a textbook for a graduate course in engineering and by researchers in the field of applied mechanics and engineers from industries who would like to further develop or apply the fast multipole BEM to solve large-scale engineering problems in their own field. This book is based on the lecture notes developed by the author over the years for a graduate course on the BEM in the Department of Mechanical Engineering at the University of Cincinnati. Many of the results are also from the research work of the author's group at Cincinnati and from the collaborative research conducted by the author with other researchers during the last 20 years.

The book is divided into six chapters. Chapter 1 is a brief introduction to the BEM and the fast multipole method. Discussions on the advantages of the BEM are highlighted.

Stokes flows are incompressible flows at low Reynolds' number [91], which can be found in many applications such as creeping flows in biological systems and fluid–structure interactions in MEMSs. Stokes flow problems were formulated with BIEs and solved by the BEM for decades with either direct or indirect BIE formulations (see, e.g., Refs. [92, 93]).

For Stokes flow problems using the fast multipole BEM, there are several approaches reported in the literature. Greengard et al. [68] developed a fast multipole formulation for directly solving the biharmonic equations in 2D elasticity with the Stokes flow as a special case. Gomez and Power [37] studied 2D cavity flow governed by Stokes equations by using both direct and indirect BIEs and the FMM in which they used Taylor series expansions of the kernels in real variables directly. Mammoli and Ingber [40] applied the fast multipole BEM to study Stokes flow around cylinders in a bounded 2D domain by using direct and indirect BIEs with the kernels expanded by a Taylor series of the real variables. In the context of modeling a MEMS, Ding and Ye [94] developed a fast BEM by using the precorrected fast Fourier transform (FFT) accelerated technique for computing drag forces with 3D MEMS models with slip BCs. Frangi and co-workers [95–98] conducted extensive research by using the direct BIE formulations and the fast multipole BEM for evaluating damping forces of 3D MEMS structures.

Although the BEM has enjoyed the reputation of ease in modeling or meshing for problems with complicated geometries or in infinite domains, its efficiency in solutions has been a serious drawback for analyzing large-scale models. For example, the BEM has been limited to solving problems with only a few thousand DOFs on a PC for many years. This is because the conventional BEM, as described in the previous chapter, produces dense and nonsymmetric matrices that although smaller in sizes, require O(N2) operations for computing the coefficients and O(N3) operations for solving the system by using direct solvers (N is the number of equations of the linear system or DOFs).

In the mid-1980s, Rokhlin and Greengard [33–35] pioneered the innovative fast multipole method (FMM) that can be used to accelerate the solutions of BEM by severalfold, promising to reduce the CPU time in FMM-accelerated BEM to O(N). With the help of the FMM, the BEM can now solve large-scale problems that are beyond the reach of other methods. We call the fast multipole accelerated BEM fast multipole BEM or simply fast BEM from now on to distinguish it from the conventional BEM described in the previous chapter. Some of the early work on fast multipole BEMs in mechanics can be found in Refs. [36–40], which show the great promise of the fast multipole BEM for solving large-scale problems. A comprehensive review of the fast multipole BIE/BEM research up to 2002 can be found in Ref. [41].

Solving acoustic wave problems is one of the most important applications of the BEM, which can be used to predict sound fields for noise control in automobiles, airplanes, and many other consumer products. Acoustic waves often exist in an infinite medium outside a structure that is in vibration (a radiation problem) or impinged on by an incident wave (a scattering problem). With the BEM, only the boundary of the structure needs to be discretized. In addition, the BCs at infinity can be taken into account analytically in the BIE formulations, and thus these conditions are satisfied exactly. The governing equation for acoustic wave problems is the Helmholtz equation, which was solved using the BIE and BEM for more than four decades (see, e.g., some of the early work in Refs. [107–120]). Especially, the work by Burton and Miller in Ref. [108] is regarded as classical work that provides a very elegant way to overcome the so-called fictitious frequency difficulties existing in the conventional BIE for exterior acoustic wave problems. Burton and Miller's BIE formulation has been used by many others in their research on the BEM for acoustic problems (see, e.g., Refs. [50, 51, 121–125]).

The development of the fast multipole BEM for solving large-scale acoustic wave problems is perhaps the most important advance in the BEM that has made the BEM unmatched by other methods in modeling acoustic wave problems.