The fast multipole method (FMM), pioneered by Rokhlin and Greengard in the mid of 1980's, can be employed to dramatically accelerate the solution of a BEM system of equations Ax = b, in which matrix A is in general dense and non-symmetrical. The main idea of the fast multipole BEM is to employ iterative solvers (such as GMRES) to solve the BEM system of equations and employ the FMM to accelerate the matrix-vector multiplication (Ax) in each iteration step, without ever forming the matrix A explicitly. In the fast multipole BEM, the node-to-node interactions in the conventional BEM are replaced by cell-to-cell interactions using a hierarchical tree structure of cells containing groups of elements. This is possible by introducing the multipole and local expansions of the kernels and employing certain translations. For more information about the fast multipole BEM, please read a comprehensive review: N. Nishimura, "Fast multipole accelerated boundary integral equation methods," Appl. Mech. Rev., 55, 299-324 (2002); or the first textbook: Y. J. Liu, Fast Multipole Boundary Element Method - Theory and Applications in Engineering, Cambridge University Press, Cambridge (2009).

Some Unique Applications of the Fast Multipole BEM Software

(Click on the pictures to see larger images)

FastBEM Software Packages for Download

Packages Updated on December 14, 2023: All packages are being updated to 64-bit programs with improved computational efficiencies.

The following fast multipole boundary element method (FastBEM) software packages (for Windows® OS only) are provided for free download and non-commercial use for the sole purpose of promoting the education, research and further development of the fast multipole BEM. Bug reports of the software and suggestions for improvements are most welcome. If you wish to collaborate and develop new capabilities for the fast multipole BEM applications, please contact Dr. Liu. See also the Copyright Statement.

Program	Description and References	Download	Examples
A1. FastBEM 2-D Potential	A fast multipole boundary element code for solving general 2-D potential problems governed by the Laplace equation, including thermal and electrostatic problems, using the dual BIE formulation (α CBIE + β HBIE). References: Chapter 3 of Ref. [1], and Refs. [2-3].	Package A1   Source code used in Refs. [1] and [2]	Porous material and MEMS
A2. FastBEM 3-D Potential	A fast multipole boundary element code for solving general 3-D potential problems governed by the Laplace equation, including thermal and electrostatic problems, using the dual BIE formulation (α CBIE + β HBIE). References: Chapter 3 of Ref. [1], and Refs. [4-5, 15].	Package A2   ANSYS to FastBEM translator code	Heat conduction and electrostatics
B1. FastBEM 2-D Elasticity	A fast multipole boundary element code for solving general 2-D linear elasticity problems with homogeneous and isotropic materials, using the dual BIE formulation (α CBIE + β HBIE). References: Chapter 4 of Ref. [1], and Refs. [6-7].	Package B1	Porous and honeycomb materials
B2. FastBEM 3-D Elasticity	A fast BEM code for solving general 3-D linear elasticity problems with homogeneous and isotropic materials, which is accelerated using the FMM, ACA and fast direct solvers. References: Chapter 4 of Ref. [1], and Refs. [8-10].	Package B2   ANSYS to FastBEM translator code	Composites and scaffold materials
C1. FastBEM 2-D Stokes Flow	A fast multipole boundary element code for solving general 2-D Stokes flow problems using the dual direct BIE formulation (α CBIE + β HBIE). References: Chapter 5 of Ref. [1], and Ref. [11].	Package C1	2-D Stokes flows
C2. FastBEM 3-D Stokes Flow	A fast multipole boundary element code for solving general 3-D Stokes flow problems using the direct BIE formulation. References: Chapter 5 of Ref. [1].	Package C2	3-D Stokes flows
D1. FastBEM 2-D Acoustics	An adaptive fast multipole boundary element code for solving general 2-D acoustic wave problems governed by the Helmholtz equation using the dual BIE formulation (α CBIE + β HBIE). References: Chapter 6 of Ref. [1], and Ref. [12].	Package D1	2-D radiation and scattering
D2. FastBEM 3-D Acoustics	An adaptive fast multipole boundary element code for solving general 3-D acoustic wave problems governed by the Helmholtz equation using the dual BIE formulation (α CBIE + β HBIE). References: Chapter 6 of Ref. [1], and Refs. [12-15].	Visit www.fastbem.com to download the commercial program	3-D radiation and scattering
E1. FastBEM 2-D Fracture	A fast BEM code for solving general 2-D linear elastic fracture mechanics problems to compute the stress intensity factors and propagation paths of multiple cracks, using the FMM, ACA and fast direct BEM solvers. Reference: Chapter 4 of Ref. [1], and Ref. [16].	Package E1	2-D radiation and scattering
References: Y. J. Liu, Fast Multipole Boundary Element Method - Theory and Applications in Engineering, Cambridge University Press, Cambridge (2009). Y. J. Liu and N. Nishimura, "The fast multipole boundary element method for potential problems: a tutorial,"  Engineering Analysis with Boundary Elements, 30, No. 5, 371-381 (2006). (Corrected Figures 4 and 5) Y. J. Liu, "Dual BIE approaches for modeling electrostatic MEMS problems with thin beams and accelerated by the fast multipole method,"  Engineering Analysis with Boundary Elements, 30, No. 11, 940-948 (2006). L. Shen and Y. J. Liu, "An adaptive fast multipole boundary element method for three-dimensional potential problems,"  Computational Mechanics,  39, No. 6, 681-691 (2007). Y. J. Liu and L. Shen, "A dual BIE approach for large-scale modeling of 3-D electrostatic problems with the fast multipole boundary element method,"  International Journal for Numerical Methods in Engineering, 71, No. 7, 837–855, (2007). Y. J. Liu, "A new fast multipole boundary element method for solving large-scale two-dimensional elastostatic problems,"  International Journal for Numerical Methods in Engineering, 65, No. 6, 863-881 (2006). Y. J. Liu, "A fast multipole boundary element method for 2-D multi-domain elastostatic problems based on a dual BIE formulation,"  Computational Mechanics, 42, No. 5, 761-773 (2008). Y. J. Liu, N. Nishimura, Y. Otani, T. Takahashi, X. L. Chen and H. Munakata, "A fast boundary element method for the analysis of fiber-reinforced composites based on a rigid-inclusion model," ASME Journal of Applied Mechanics, 72, No. 1, 115-128 (2005). Y. J. Liu, N. Nishimura and Y. Otani, "Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method,"  Computational Materials Science, 34, No. 2, 173-187 (2005). Y. J. Liu, N. Nishimura, D. Qian, N. Adachi, Y. Otani and V. Mokashi, "A boundary element method for the analysis of CNT/polymer composites with a cohesive interface model based on molecular dynamics,"  Engineering Analysis with Boundary Elements, 32, No. 4, 299–308 (2008). Y. J. Liu, "A new fast multipole boundary element method for solving 2-D Stokes flow problems based on a dual BIE formulation,"  Engineering Analysis with Boundary Elements, 32, No. 2, 139-151 (2008). Y. J. Liu, L. Shen and M. Bapat, "Development of the Fast Multipole Boundary Element Method for Acoustic Wave Problems," in: Recent Advances in the Boundary Element Methods, edited by G. Manolis and D. Polyzos (Springer-Verlag, Berlin, 2009). L. Shen and Y. J. Liu, "An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton-Miller formulation,"  Computational Mechanics, 40, No. 3, 461-472 (2007). M. S. Bapat, L. Shen and Y. J. Liu, "Adaptive fast multipole boundary element method for three-dimensional half-space acoustic wave problems,"  Engineering Analysis with Boundary Elements, 33, Nos. 8-9, 1113-1123 (2009). M. S. Bapat and Y. J. Liu, "A new adaptive algorithm for the fast multipole boundary element method,"  CMES: Computer Modeling in Engineering & Sciences, 58, No. 2, 161-184 (2010). Y. J. Liu, Y. X. Li, and W. Xie, "Modeling of multiple crack propagation in 2-D elastic solids by the fast multipole boundary element method," Engineering Fracture Mechanics, 172, 1-16 (2017). © 2004-2023. Copyright Notice and Disclaimers: The above fast multipole boundary element method (FastBEM) software packages are copyrighted materials of the authors. No part of the packages, either the executable or the source codes, can be used for any commercial applications and distributions without prior written permissions of the original authors. Proper acknowledgment should be given in publications resulting from the use of these software. The authors retain all the rights of these software. There is no warranty, expressed or implied, for the use of these software. The authors are not responsible for any possible damages in using these software and no technical support is available to users of these software.

Dr. Yijun Liu

Web:  www.yijunliu.com