Fast Multipole Boundary Element Method (FastBEM) Softwarefor Education, Research and Further DevelopmentDr. Yijun Liu, CAE Research Lab, University of Cincinnati, Ohio, USA

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 nonsymmetrical. 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 matrixvector multiplication (Ax) in each iteration step, without ever forming the matrix A explicitly. In the fast multipole BEM, the nodetonode interactions in the conventional BEM are replaced by celltocell 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, 299324 (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)
A. Thermal Analysis: Fuel Cells (There are 9,000 small side holes in this model! Total DOFs = 530,000, solved on a desktop PC) 
B. Elasticity: Fiber Composites (Up to 16,000 CNT fibers and total DOFs = 28,800,000, solved on a supercomputer at Kyoto University) 
C. Stokes Flow: MEMS (This is an exterior Stokes flow problem. Total DOFs = 1,087,986, solved on a desktop PC) 
D. Acoustics: Noise Control (This is an exterior acoustic wave radiation problem. Total complex DOFs = 541,152, solved on a desktop PC) 
FastBEM Software Packages for Download
First Release (January 18, 2017): The new package E1. FastBEM 2D Fracture is released, which can be applied to solve general 2D linear elastic fracture mechanics problems to compute the stress intensity factors and/or propagation paths of multiple cracks, using the FMM, ACA and fast direct BEM solvers.
Updated (January 18, 2017): FastBEM 3D Elasticity package is updated, which is based on a new FMM, ACA, and fast direct BEM solvers. An ANSYS to FastBEM translator code is also available for building 3D elastostatic BEM models using ANSYS.
The following fast multipole boundary element method (FastBEM) software packages (for Windows® OS only) are provided for free download and noncommercial 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
2D Potential
A fast multipole boundary element code for solving general 2D 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. [23].
Porous material and MEMS
A2. FastBEM
3D Potential
A fast multipole boundary element code for solving general 3D 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. [45, 15].
Heat conduction and electrostatics
B1. FastBEM
2D Elasticity
A fast multipole boundary element code for solving general 2D 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. [67].
Porous and honeycomb materials
B2. FastBEM
3D Elasticity
A fast BEM code for solving general 3D 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. [810].
Composites and scaffold materials
C1. FastBEM
2D Stokes Flow
A fast multipole boundary element code for solving general 2D Stokes flow problems using the dual direct BIE formulation (α CBIE + β HBIE).
References: Chapter 5 of Ref. [1], and Ref. [11].
2D Stokes flows
C2. FastBEM
3D Stokes Flow
A fast multipole boundary element code for solving general 3D Stokes flow problems using the direct BIE formulation.
References: Chapter 5 of Ref. [1].
3D Stokes flows
D1. FastBEM
2D Acoustics
An adaptive fast multipole boundary element code for solving general 2D 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].
2D radiation and scattering
D2. FastBEM
3D Acoustics
An adaptive fast multipole boundary element code for solving general 3D 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. [1215].
Visit www.fastbem.com to download the commercial program
3D radiation and scattering
E1. FastBEM
2D Fracture
A fast BEM code for solving general 2D 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].
2D 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,
371381 (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, 940948 (2006).
L. Shen and Y. J. Liu, "An adaptive fast multipole boundary element method for threedimensional potential problems," Computational Mechanics, 39, No. 6, 681691 (2007).
Y. J. Liu and L. Shen, "A dual BIE approach for largescale modeling of 3D 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 largescale twodimensional elastostatic problems," International Journal for Numerical Methods in Engineering, 65, No. 6, 863881 (2006).
Y. J. Liu, "A fast multipole boundary element method for 2D multidomain elastostatic problems based on a dual BIE formulation," Computational Mechanics, 42, No. 5, 761773 (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 fiberreinforced composites based on a rigidinclusion model," ASME Journal of Applied Mechanics, 72, No. 1, 115128 (2005).
Y. J. Liu, N. Nishimura and Y. Otani, "Largescale modeling of carbonnanotube composites by a fast multipole boundary element method," Computational Materials Science, 34, No. 2, 173187 (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 2D Stokes flow problems based on a dual BIE formulation," Engineering Analysis with Boundary Elements, 32, No. 2, 139151 (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 (SpringerVerlag, Berlin, 2009).
L. Shen and Y. J. Liu, "An adaptive fast multipole boundary element method for threedimensional acoustic wave problems based on the BurtonMiller formulation," Computational Mechanics, 40, No. 3, 461472 (2007).
M. S. Bapat, L. Shen and Y. J. Liu, "Adaptive fast multipole boundary element method for threedimensional halfspace acoustic wave problems," Engineering Analysis with Boundary Elements, 33, Nos. 89, 11131123 (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, 161184 (2010).
Y. J. Liu, Y. X. Li, and W. Xie, "Modeling of multiple crack propagation in 2D elastic solids by the fast multipole boundary element method," Engineering Fracture Mechanics, 172, 116 (2017).
© 20042017. 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.
Mechanical Engineering
University of Cincinnati, P.O. Box 210072
Cincinnati, OH 452210072, U.S.A.
Tel.: (513) 5564607 (Voice), (513) 5563390 (Fax)
Email: Yijun.Liu@uc.edu
Web: www.yijunliu.com
Site accessed times since January 15, 2007.  © 19962017 Yijun Liu, University of Cincinnati 
Last updated: January 18, 2017. 