Solving Boundary Integral Problems with BEM++

Abstract

Many important partial differential equation problems in homogeneous media, such as those of acoustic or electromagnetic wave propagation, can be represented in the form of integral equations on the boundary of the domain of interest. In order to solve such problems, the boundary element method (BEM) can be applied. The advantage compared to domain-discretisation-based methods such as finite element methods is that only a discretisation of the boundary is necessary, which significantly reduces the number of unknowns. Yet, BEM formulations are much more difficult to implement than finite element methods. In this article, we present BEM++, a novel open-source library for the solution of boundary integral equations for Laplace, Helmholtz and Maxwell problems in three space dimensions. BEM++ is a C++ library with Python bindings for all important features, making it possible to integrate the library into other C++ projects or to use it directly via Python scripts. The internal structure and design decisions for BEM++ are discussed. Several examples are presented to demonstrate the performance of the library for larger problems.

References

1. F. Andriulli, K. Cools, H. Bagci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen. 2008. A multiplicative Calderon preconditioner for the electric field integral equation. IEEE Trans. Antenn. Propagat. 56, 8, 2398--2412.
2. R. A. Bartlett. 2007. Thyra linear operators and vectors. Tech. Rep. SAND2007-5984. Sandia National Laboratories, Albuquerque, NM.
3. P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, R. Kornhuber, M. Ohlberger, and O. Sander. 2008a. A generic grid interface for parallel and adaptive scientific computing. Part II: Implementation and tests in DUNE. Computing 82, 2--3, 121--138.
4. P. Bastian, M. Blatt, A. Dedner, C. Engwer, R. Klöfkorn, M. Ohlberger, and O. Sander. 2008b. A generic grid interface for parallel and adaptive scientific computing. Part I: Abstract framework. Computing 82, 2--3, 103--119.
5. D. Beazley. 2003. Automated scientific software scripting with SWIG. Future Gen. Comput. Syst. 19, 5, 599--609. (Tools for Program Development and Analysis. Best papers from two Technical Sessions, at ICCS2001, San Francisco, CA, USA, and ICCS2002, Amsterdam, The Netherlands.)
6. M. Bebendorf. 2008. Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems. Lecture Notes in computational Science and Engineering, vol. 63, Springer, Berlin Heidelberg.
7. M. Bebendorf. 2012. Another software library on hierarchical matrices for elliptic differential equations (AHMED). http://bebendorf.ins.uni-bonn.de/AHMED.html.
8. T. Betcke, S. Arridge, J. Phillips, M. Schweiger, and W. Śmigaj. 2013. Solution of electromagnetic problems with BEM++. In Oberwolfach Report, Vol. 03/2013.
9. C. J. Bouwkamp. 1950. On Bethe's theory of diffraction by small holes. Philips Res. Rep. 5, 321.
10. A. Buffa and R. Hiptmair. 2003. Galerkin boundary element methods for electromagnetic scattering. In Topics in Computational Wave Propagation, Springer, 83--124.
11. A. J. Burton and G. F. Miller. 1971. The application of integral equation methods to the numerical solution of some exterior boundary-value problems. R. Soc. London Proc. Ser. A 323, 201--210.
12. H. Cheng, W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J. F. Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao. 2006. A wideband fast multipole method for the Helmholtz equation in three dimensions. J. Comp. Phys. 216, 1, 300--325.
13. A. Chernov and C. Schwab. 2012. Exponential convergence of Gauss--Jacobi quadratures for singular integrals over simplices in arbitrary dimension. SIAM J. Numer. Anal. 50, 3, 1433--1455.
14. A. Chernov, T. von Petersdorff, and C. Schwab. 2011. Exponential convergence of hp quadrature for integral operators with Gevrey kernels. ESAIM: Math. Model. Numer. Analy. 45, 3, 387--422.
15. D. L. Colton and R. Kress. 2013. Inverse Acoustic and Electromagnetic Scattering Theory. Springer.
16. D. A. Dunavant. 1985. High degree efficient symmetrical Gaussian quadrature rules for the triangle. Int. J. Num. Meth. Eng. 21, 6, 1129--1148.
17. DUNE. 2012. Distributed and Unified Numerics Environment (DUNE). http://www.dune-project.org.
18. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff. 1998. Extraordinary optical transmission through sub-wavelength hole arrays. Nature 391, 667--669.
19. H. Estrada, P. Candelas, A. Uris, F. Belmar, F. J. García de Abajo, and F. Meseguer. 2008. Extraordinary sound screening in perforated plates. Phys. Rev. Lett. 101, 8, 084302.
20. F. J. García de Abajo. 2007. Colloquium: Light scattering by particle and hole arrays. Rev. Mod. Phys. 79, 1267--1290.
21. C. Geuzaine and J.-F. Remacle. 2009. GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Num. Meth. Engng 79, 11, 1309--1331.
22. C. Geuzaine and J.-F. Remacle. 2012. GMSH. http://geuz.org/gmsh.
23. C. Gräser and O. Sander. 2012. Dune-FoamGrid. http://users.dune-project.org/projects/dune-foamgrid.
24. D. J. Griffiths. 1998. Introduction to Electrodynamics 3rd Ed. Benjamin Cummings.
25. H. Harbrecht and R. Schneider. 2006. Wavelet Galerkin schemes for boundary integral equations—Implementation and quadrature. SIAM J. Sci. Comput. 27, 4, 1347--1370.
26. R. F. Harrington and J. L. Harrington. 1996. Field Computation by Moment Methods. Oxford University Press.
27. M. A. Heroux, R. A. Bartlett, V. E. Howle, R. J. Hoekstra, J. J. Hu, T. G. Kolda, R. B. Lehoucq, K. R. Long, R. P. Pawlowski, E. T. Phipps, A. G. Salinger, H. K. Thornquist, R. S. Tuminaro, J. M. Willenbring, A. Williams, and K. S. Stanley. 2005. An overview of the Trilinos project. ACM Trans. Math. Softw. 31, 3, 397--423.
28. R. Hiptmair and L. Kielhorn. 2012. BETL — A generic boundary element template library. Tech. Rep. 2012-36. Seminar for Applied Mathematics, ETH Zürich.
29. Intel. 2012. Intel Threading Building Blocks. http://threadingbuildingblocks.org.
30. L. Kielhorn. 2012. Boundary Element Template Library (BETL). http://www.sam.math.ethz.ch/betl.
31. Kitware. 2012a. Paraview. http://www.paraview.org.
32. Kitware. 2012b. Visualizaton Toolkit (VTK). http://www.vtk.org.
33. R. Kleinman and P. Martin. 1988. On single integral equations for the transmission problem of acoustics. SIAM J. Appl. Math. 48, 2, 307--325.
34. R. Kress. 1985. Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering. Quart. J. Mech. Appl. Math. 38, 2, 323--341.
35. M. Lenoir and N. Salles. 2012. Evaluation of 3-d singular and nearly singular integrals in Galerkin BEM for thin layers. SIAM J. Sci. Comput. 34, 6, A3057--A3078.
36. H. Liu and P. Lalanne. 2008. Microscopic theory of the extraordinary optical transmission. Nature 452, 728--731.
37. M. Maischak. 2013. MaiProgs. http://www.ifam.uni-hannover.de/~maiprogs.
38. M. Messner, M. Messner, P. Urthaler, and F. Rammerstorfer. 2010. Hyperbolic and Elliptic Numerical Analysis (HyENA). http://portal.tugraz.at/portal/page/portal/Files/i2610/files/Forschung/Software/HyENA/html/index.html.
39. J.-C. Nédélec. 2001. Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems. Springer.
40. G. Of. 2008. An efficient algebraic multigrid preconditioner for a fast multipole boundary element method. Computing 82, 2--3, 139--155.
41. G. Of, O. Steinbach, and W. L. Wendland. 2006. The fast multipole method for the symmetric boundary integral formulation. Ima J. Numer. Anal. 26, 2, 272--296.
42. OpenCASCADE. 2012. OpenCASCADE: Shape gallery. http://www.opencascade.org/showroom/shapegallery.
43. A. G. Polimeridis and J. R. Mosig. 2010. Complete semi-analytical treatment of weakly singular integrals on planar triangles via the direct evaluation method. Intl. J. Numer. Meth. Eng. 83, 1625--1650.
44. A. G. Polimeridis, F. Vipiana, J. R. Mosig, and D. R. Wilton. 2013. DIRECTFN: Fully numerical algorithms for high precision computation of singular integrals in Galerkin SIE methods. IEEE Trans. Anten. Propag. 61, 3112--3122.
45. P. Ramachandran. 2004--2005. An introduction to Traited VTK (TVTK). http://docs.enthought.com/mayavi/tvtk/README.html.
46. P.-A. Raviart and J.-M. Thomas. 1977. A mixed finite element method for 2nd order elliptic problems. In Mathematical Aspects of Finite Element Methods, Springer, 292--315.
47. S. Rjasanow and O. Steinbach. 2007. The Fast Solution of Boundary Integral Equations. Springer, Berlin Heidelberg.
48. C. Sanderson. 2012. Armadillo: C++ linear algebra library. http://arma.sourceforge.net.
49. S. A. Sauter and C. Schwab. 2011. Boundary Element Methods. Springer Series in Computational Mathematics, 39, Springer, Berlin Heidelberg.
50. K. Schmidt. 2013. Concepts -- A numerical C++ library. http://www.concepts.math.ethz.ch.
51. W. Śmigaj, T. Betcke, S. Arridge, J. Phillips, and M. Schweiger. 2013. Solving boundary integral problems with BEM++. http://www.bempp.org/files/bempp-toms-preprint.pdf.
52. P. Šolín. 2005. Partial Differential Equations and the Finite Element Method. Wiley.
53. O. Steinbach. 2008. Numerical Approximation Methods for Elliptic Boundary Value Problems. Springer.
54. SWIG. 2012. Simplified Wrapper and Interface Generator (SWIG). http://www.swig.org.
55. B. A. Szabo and I. Babuška. 1991. Finite Element Analysis. Wiley.
56. Trilinos. 2012. Trilinos. http://trilinos.sandia.gov.
57. I. van den Bosch. 2013. Puma-EM. http://puma-em.sourceforge.net.
58. P. Wieleba and J. Sikora. 2011. “BEMLAB”—universal, open source, boundary element method library applied in micro-electro-mechanical systems. Studies Appl. Electromagn. Mech. 35, 173--182.

Index Terms

• Mathematics of computing

  • Mathematical software