volume | PIER Journals

Abstract

We describe a technique to analytically compute the multipole moments of a charge distribution confined to a planar triangle, which may be useful in solving the Laplace equation using the fast multipole boundary element method (FMBEM) and for charged particle tracking. This algorithm proceeds by performing the necessary integration recursively within a specific coordinate system, and then transforming the moments into the global coordinate system through the application of rotation and translation operators. This method has been implemented and found use in conjunction with a simple piecewise constant collocation scheme, but is generalizable to non-uniform charge densities. When applied to low aspect ratio (≤100) triangles and expansions with degree up to 32, it is accurate and efficient compared to simple two-dimensional Gauss-Legendre quadrature.

1. Poljak, D. and C. A. Brebbia, Boundary Element Methods for Electrical Engineers, Vol. 4, WIT Press, 2005.

2. Szilagyi, M., Electron and Ion Optics, Springer, 1988.
doi:10.1007/978-1-4613-0923-9

3. Liu, Y., Fast Multipole Boundary Element Method: Theory and Applications in Engineering, Cambridge University Press, 2009.
doi:10.1017/CBO9780511605345

4. Lazić, P., H. Štefančić, and H. Abraham, "The robin hood method --- A new view on differential equations," Engineering Analysis with Boundary Elements, Vol. 32, No. 1, 76-89, 2008.
doi:10.1016/j.enganabound.2007.06.004

5. Formaggio, J. A., P. Lazić, T. J. Corona, H. Štefančić, H. Abraham, and F. Glück, "Solving for micro- and macro-scale electrostatic configurations using the robin hood algorithm," Progress In Electromagnetics Research B, Vol. 39, 1-37, 2012.
doi:10.2528/PIERB11112106

6. Rokhlin, V., "Rapid solution of integral equations of classical potential theory," Journal of Computational Physics, Vol. 60, No. 2, 187-207, 1985.
doi:10.1016/0021-9991(85)90002-6

7. Greengard, L. and V. Rokhlin, "The rapid evaluation of potential fields in three dimensions," Vortex Methods, 121-141, 1988.
doi:10.1007/BFb0089775

8. Beatson, R. and L. Greengard, "A short course on fast multipole methods," Wavelets, Multilevel Methods and Elliptic PDEs, 1-37, Oxford University Press, 1997.

9. Epton, M. A. and B. Dembart, "Multipole translation theory for the three-dimensional laplace and Helmholtz equations," SIAM Journal on Scientific Computing, Vol. 16, No. 4, 865-897, 1995.
doi:10.1137/0916051

10. Van Gelderen, M., "The shift operators and translations of spherical harmonics," DEOS Progress Letters, Vol. 98, 57, 1998.

11. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, 1998.

12. Lether, F. G., "Computation of double integrals over a triangle," Journal of Computational and Applied Mathematics, Vol. 2, No. 3, 219-224, 1976.
doi:10.1016/0771-050X(76)90008-5

13. Mousa, M.-H., R. Chaine, S. Akkouche, and E. Galin, "Toward an efficient triangle-based spherical harmonics representation of 3D objects," Computer Aided Geometric Design, Vol. 25, No. 8, 561-575, 2008.
doi:10.1016/j.cagd.2008.06.004

14. Proriol, J., "Sur une famille de polynomes á deux variables orthogonaux dans un triangle," CR Acad. Sci. Paris, Vol. 245, 2459-2461, 1957.

15. Dubiner, M., "Spectral methods on triangles and other domains," Journal of Scientific Computing, Vol. 6, No. 4, 345-390, 1991.
doi:10.1007/BF01060030

16. Owens, R., "Spectral approximations on the triangle," Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 454, No. 1971, 857-872, 1998.
doi:10.1098/rspa.1998.0189

17. Koornwinder, T., "Two-variable analogues of the classical orthogonal polynomials," Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, University Wisconsin, Madison, Wis., 1975), 435-495, Academic Press, New York, 1975.

18. Wait, R. and A. Mitchell, Finite Element Analysis and Applications, Books on Demand, 1985.

19. Taylor, R. L., "On completeness of shape functions for finite element analysis," International Journal for Numerical Methods in Engineering, Vol. 4, No. 1, 17-22, 1972.
doi:10.1002/nme.1620040105

20. Barnhill, R. E. and J. A. Gregory, "Polynomial interpolation to boundary data on triangles," Mathematics of Computation, Vol. 29, No. 131, 726-735, 1975.
doi:10.1090/S0025-5718-1975-0375735-3

21. Chen, G. and J. Zhou, Boundary Element Methods, Computational Mathematics and Applications, Academic Press, 1992.

22. Gander, W., "Change of basis in polynomial interpolation," Numerical Linear Algebra with Applications, Vol. 12, No. 8, 769-778, 2005.
doi:10.1002/nla.450

23. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Courier Dover Publications, 1966.

24. Mason, J. C. and D. C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC, 2002.
doi:10.1201/9781420036114

25. Pinchon, D. and P. E. Hoggan, "Rotation matrices for real spherical harmonics: General rotations of atomic orbitals in space-fixed axes," Journal of Physics A: Mathematical and Theoretical, Vol. 40, No. 7, 1597, 2007.
doi:10.1088/1751-8113/40/7/011

26. Gimbutas, Z. and L. Greengard, "A fast and stable method for rotating spherical harmonic expansions," Journal of Computational Physics, Vol. 228, No. 16, 5621-5627, 2009.
doi:10.1016/j.jcp.2009.05.014

27. Wigner, E. and J. J. Griffin, Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959.

28. Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton University Press, 1958.

29. Choi, C. H., J. Ivanic, M. S. Gordon, and K. Ruedenberg, "Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion," The Journal of Chemical Physics, Vol. 111, No. 19, 8825-8831, 1999.
doi:10.1063/1.480229

30. Lessig, C., T. De Witt, and E. Fiume, "Efficient and accurate rotation of finite spherical harmonics expansions," Journal of Computational Physics, Vol. 231, No. 2, 243-250, 2012.
doi:10.1016/j.jcp.2011.09.014

31. White, C. A. and M. Head-Gordon, "Rotating around the quartic angular momentum barrier in fast multipole method calculations," The Journal of Chemical Physics, Vol. 105, 5061, 1996.
doi:10.1063/1.472369

32. Berntsen, J., T. O. Espelid, and A. Genz, "An adaptive algorithm for the approximate calculation of multiple integrals," ACM Transactions on Mathematical Software, Vol. 17, 437-451, Dec. 1991.
doi:10.1145/210232.210233

33. Cowper, G., "Gaussian quadrature formulas for triangles," International Journal for Numerical Methods in Engineering, Vol. 7, No. 3, 405-408, 1973.
doi:10.1002/nme.1620070316

34. Duffy, M. G., "Quadrature over a pyramid or cube of integrands with a singularity at a vertex," SIAM Journal on Numerical Analysis, Vol. 19, No. 6, 1260-1262, 1982.
doi:10.1137/0719090

35. Golub, G. H. and J. H. Welsch, "Calculation of gauss quadrature rules," Mathematics of Computation, Vol. 23, No. 106, 221-230, 1969.
doi:10.1090/S0025-5718-69-99647-1

36. Saad, Y. and M. H. Schultz, "GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems," SIAM Journal on Scientific and Statistical Computing, Vol. 7, No. 3, 856-869, 1986.
doi:10.1137/0907058

37. Read, F., "Improved extrapolation technique in the boundary element method to find the capacitances of the unit square and cube," Journal of Computational Physics, Vol. 133, No. 1, 1-5, 1997.
doi:10.1006/jcph.1996.5519

38. Hudson, R. G. and J. Lipka, A Table of Integrals, John Wiley & Sons, 1917.

39. Peirce, B. O., A Short Table of Integrals, Ginn & Company, 1910.

40. Papantonopoulou, A., Algebra: Pure & Applied, Prentice Hall, 2002.

41. Beachy, J. A. and W. D. Blair, Abstract Algebra, Waveland Press, 2006.

Mr. John Barrett

Prof. Joseph A. Formaggio

Dr. Thomas Joseph Corona