It is a widely held view that analytical integration is more accurate than the numerical one. In some special cases, however, numerical integration can be more advantageous than analytical integration. In our paper we show this benefit for the case of electric potential and field computation of charged triangles and rectangles applied in the boundary element method (BEM). Analytical potential and field formulas are rather complicated (even in the simplest case of constant charge densities), they have usually large computation times, and at field points far from the elements they suffer from large rounding errors. On the other hand, Gaussian cubature, which is an efficient numerical integration method, yields simple and fast potential and field formulas that are very accurate far from the elements. The simplicity of the method is demonstrated by the physical picture: the triangles and rectangles with their continuous charge distributions are replaced by discrete point charges, whose simple potential and field formulas explain the higher accuracy and speed of this method. We implemented the Gaussian cubature method for the purpose of BEM computations both with CPU and GPU, and we compare its performance with two different analytical integration methods. The ten different Gaussian cubature formulas presented in our paper can be used for arbitrary high-precision and fast integrations over triangles and rectangles.
2. Segerlind, L. J., "Applied Finite Element Analysis," J. Wiley & Sons, 1984.
3. Cook, R. D., et al., Concepts and Applications of Finite Element Analysis, J. Wiley & Sons, New York, 1989.
4. Gupta, O. P., Finite and Boundary Element Methods in Engineering, A. A. Balkema Publishers, Rotterdam Brookfield, 1999.
5. Kythe, P. K., An Introduction to the Boundary Element Method, CRC Press, London Tokyo, 1995.
6. Gaul, L., M. Kogl, and M. Wagner, Boundary Element Methods for Engineers and Scientists, Springer-Verlag, Berlin, 2003.
7. Beer, G., I. Smith, and Ch. Duenser, The Boundary Element Method with Programming, Springer, Wien New York, 2008.
8. Brebbia, C. A., J. C. F. Telles, and L. C. Wrobel, Boundary Element Techniques, Springer-Verlag, Berlin, 1984.
9. Brebbia, C. A. and J. Dominguez, Boundary Elements an Introductory Course, McGraw-Hill Comp., New York, 1992.
10. Corona, T. J., Tools for electromagnetic field simulation in the KATRIN experiment, Master Thesis, MIT, 2009.
11. Szilagyi, M., Electron and Ion Optics, Plenum Press, New York and London, 1988.
12. Hawkes, P. W. and E. Kasper, Principles of Electron Optics, Vol. 1, Academic Press, Harcourt Brace Jovanovich, 1989.
13. Kraus, Ch., et al., "Final results from phase II of the Mainz neutrino mass search in tritium β decay," Eur. Phys. J. C, Vol. 40, 447-468, 2005.
14. Lobashev, V. M., "The search for the neutrino mass by direct method in the tritium beta-decay and perspectives of study it in the project KATRIN," Nucl. Phys. A, Vol. 719, c153-c160, 2003.
15. Gluck, F., et al., "The neutron decay retardation spectrometer aSPECT: Electromagnetic design and systematic effects," Eur. Phys. J. A, Vol. 23, 135-146, 2005; Baessler, S., et al., ``First measurements with the neutron decay spectrometer aSPECT,” Eur. Phys. J. A, Vol. 38, 17–26, 2008.
16. Angrik, J., et al., " (KATRIN Collaboration), KATRIN Design Report 2004,", Wissenschaftliche Berichte FZKA 7090, http://bibliothek.fzk.de/zb/berichte/FZKA7090.pdf.
17. Prall, M., et al., "The KATRIN pre-spectrometer at reduced filter energy," New Journal of Physics, Vol. 14, 073054, 2012.
18. Arenz, M., et al., "Commissioning of the vacuum system of the KATRIN main spectrometer," J. Instrum., Vol. 11, P04011, 2016.
19. Corona, T. J., Methodology and application of high performance electrostatic field simulation in the KATRIN experiment, Ph.D. Thesis, Chapel Hill, 2014, https://cdr.lib.unc.edu/record/uuid:6f44a9c2-f053-404a-b726-b960d5772619.
20. Furse, D., Techniques for direct neutrino mass measurement utilizing tritium β-decay, Ph.D. Thesis, MIT, 2015.
21. Groh, S., Modeling of the response function and measurement of transmission properties of the KATRIN experiment, Ph.D. Thesis, KIT, 2015.
22. Chari, M. V. K. and S. J. Salon, Numerical Methods in Electromagnetism, Academic Press, San Diego, 2000.
23. Ueberhuber, Ch. W., "Numerical computation," Methods, Software and Analysis, Vol. 1, Springer, Berlin, 1997.
24. Goldberg, D., "What every computer scientist should know about floating-point arithmetic," ACM Computing Surveys, Vol. 23, 5-48, 1991.
25. Formaggio, J., et al., "Solving for micro- and macro-scale electrostatic configurations using the Robin Hood algorithm," Progress In Electromagnetics Research B, Vol. 39, 1-37, 2011.
26. Hanninen, I., M. Taskinen, and J. Sarvas, "Singularity subtraction integral formulae for surface integral equations with rwg, rooftop and hybrid basis functions," Progress In Electromagnetics Research, Vol. 63, 243-278, 2006.
27. Hilk, D., Electric field simulations and electric dipole investigations at the KATRIN main spectrometer, Ph.D. Thesis, KIT, 2017.
28. Rao, S. M., et al., "A simple numerical solution procedure for statics problems involving arbitraryshaped surfaces," IEEE Transactions on Antennas and Propagation, Vol. 27, 604-608, 1979.
29. Okon, E. E. and R. F. Harrington, "The potential due to a uniform source distribution over a triangular domain," International Journal for Numerical Methods in Engineering, Vol. 18, 1401-1419, 1982.
30. Davey, K. and S. Hinduja, "Analytical integration of linear three-dimensional triangular elements in BEM," Applied Mathematical Modelling, Vol. 13, 450-461, 1989.
31. Tatematsu, A., S. Hamada, and T. Takuma, "Analytical expressions of potential and electric field generated by a triangular surface charge with a high-order charge density distribution," Electrical Engineering in Japan, Vol. 139, 9-17, 2002.
32. Lopez-Pena, S., A. G. Polimeridis, and J. R. Mosig, "On the analytic-numeric treatment of weakly singular integrals on arbitrary polygonal domains," Progress In Electromagnetics Research, Vol. 117, 339-355, 2011.
33. Carley, M. J., "Analytical formulae for potential integrals on triangles," Journal of Applied Mechanics, Vol. 80, 041008, 1-7, 2013.
34. Durand, E., Electrostatique, Vol. I, Masson et Cie, 1964.
35. Eupper, M., Eine verbesserte Integralgleichungsmethode zur numerischen Losung dreidimensionaler Dirichletprobleme und ihre Anwendung in der Elektronenoptik, Ph.D. Thesis, Tubingen, 1985.
36. Birtles, A. B., B. J. Mayo, and A. W. Bennett, "Computer technique for solving 3-dimensional electron-optics and capacitance problems," Proc. IEE, Vol. 120, 213-220, 1973; Erratum: Proc. IEE, Vol. 120, 559, 1973.
37. Evans, G., Practical Numerical Integration, John Wiley & Sons, Chichester, 1993.
38. Kythe, P. K. and M. R. Schaferkotter, Handbook of Computational Methods for Integration, Chapman & Hall/CRC, London, 2005.
39. Stroud, A. H., Approximate Calculation of Multiple Integrals, Prentice Hall Inc., New Jersey, 1971.
40. Engels, H., Numerical Quadrature and Cubature, Academic Press, London, 1980.
41. Engeln-Mullges, G. and F. Uhlig, "Numerical Algorithms with C," Springer, 1996.
42. Radon, J., "Zur mechanischen Kubatur," Monatsh. Math., Vol. 52, 286-300, 1948.
43. Hammer, P. C. and A. H. Stroud, "Numerical integration over simplexes," Math. Tables Aids Comput., Vol. 10, 137-139, 1956.
44. Hammer, P. C. and A. H. Stroud, "Numerical evaluation of multiple integrals II," Math. Tables Aids Comput., Vol. 12, 272-280, 1958.
45. Hammer, P. C., O. J. Marlowe, and A. H. Stroud, "Numerical integration over simplexes and cones," Math. Tables Aids Comput., Vol. 10, 130-137, 1956.
46. Gatermann, K., "The construction of symmetric cubature formulas for the square and the triangle," Computing, Vol. 40, 229-240, 1988.
47. Lyness, J. N. and D. Jespersen, "Moderate degree symmetric quadrature rules for the triangle," J. Inst. Maths. Applics, Vol. 15, 19-32, 1975.
48. Papanicolopulos, S., "Computation of moderate-degree fully-symmetric cubature rules on the triangle using symmetric polynomials and algebraic solving," Computers and Mathematics with Applications, Vol. 69, 650-666, 2015.
49. Albrect, J. and L. Collatz, "Zur numerischen auswertung mehrdimensionaler integrale," Zeitschrift fur Angewandte Mathematik und Mechanik, Vol. 38, 1-15, 1958.
50. Tyler, G. W., "Numerical integration of functions of several variables," Canad. J. Math., Vol. 5, 393-412, 1953.
51. Moller, H. M., "Kubaturformeln mit minimaler Knotenzahl," Numer. Math., Vol. 25, 185-200, 1976.
52. Cools, R. and A. Haegemans, "Another step forward in searching for cubature formulae with a minimal number of knots for the square," Computing, Vol. 40, 139-146, 1988.
53. Rabinowitz, P. and N. Richter, "Perfectly symmetric two-dimensional integration formulas with minimal numbers of points," Mathematics of Computation, Vol. 23, 765-779, 1969.
54. Omelyan, I. P. and V. B. Solovyan, "Improved cubature formulae of high degrees of exactness for the square," Journal of Computational and Applied Mathematics, Vol. 188, 190-204, 2006.
55. Dunavant, D. A., "High degree efficient symmetrical gaussian quadrature rules for the triangle," International Journal for Numerical Methods in Engineering, Vol. 21, 1129-1148, 1985.
56. Haegemans, A. and R. Piessens, "Construction of cubature formulas of degree eleven for symmetric planar regions, using orthogonal polynomials," Numer. Math., Vol. 25, 139-148, 1976.
57. Haegemans, A. and R. Piessens, "Construction of cubature formulas of degree seven and nine symmetric planar regions, using orthogonal polynomials," SIAM Journal on Numerical Analysis, Vol. 14, 492-508, 1977.
58. Wandzura, S. and H. Xiao, "Symmetric quadrature rules on a triangle," Computers and Mathematics with Applications, Vol. 45, 1829-1840, 2003.
59. Zhang, L., T. Cui, and H. Liu, "A set of symmetric quadrature rules on triangles and tetrahedra," Journal of Computational Mathematics, Vol. 27, 89-96, 2009.
60. Cools, R. and P. Rabinowitz, "Monomial cubature rules since Stroud: A compilation," Journal of Computational and Applied Mathematics, Vol. 48, 309-326, 1993.
61. Lyness, J. N. and R. Cools, "A survey of numerical cubature over triangles," Proceedings of Symposia in Applied Mathematics, 127-150, Preprint MCS-P410-0194, Argonne National Laboratory, 1994.
62. Cools, R., "Monomial cubature rules since Stroud: A compilation — Part 2," Journal of Computational and Applied Mathematics, Vol. 112, 21-27, 1999.
63. Cools, R., "Advances in multidimensional integration," Journal of Computational and Applied Mathematics, Vol. 149, 1-12, 2002.
64. Cools, R., "An encyclopaedia of cubature formulas," Journal of Complexity, Vol. 19, 445-453, 2003.
65. Cools, R., Encyclopaedia of Cubature Formulas, web page: http://nines.cs.kuleuven.be/ecf/.
66. Duffy, M. G., "Quadrature over a pyramid or cube of integrands with a singularity at a vertex," SIAM J. Numer. Anal., Vol. 19, 1260-1262, 1982.
67. Valerius, K., "The wire electrode system for the KATRIN main spectrometer," Prog. Part. Nucl. Phys., Vol. 64, 291-293, 2010.
68. Stern, S., Untersuchung der Untergrundeigenschaften des KATRIN Hauptspektrometers mit gepulsten elektrischen Dipolfeldern, Bachelor Thesis, KIT, 2016.
69. Combe, R., Design optimization of the KATRIN transport section and investigation of related background contribution,, Master Thesis, KIT, 2015.
70. Barrett, J., A spatially resolved study of the KATRIN main spectrometer using a novel fast multipole method, Ph.D. Thesis, MIT, 2017.
71. Gosda, W., A new preconditioning approach using the fast fourier transformation on multipoles, Diploma Thesis, KIT, 2015.
72. Barrett, J., J. Formaggio, and T. J. Corona, "A method to calculate the spherical multipole expansion of the electrostatic charge distribution on a triangular boundary element," Progress In Electromagnetics Research B, Vol. 63, 123-143, 2015.
73. Lazic, P. H., H. Stefancic, and H. Abraham, "The robin hood method — A novel numerical method for electrostatic problems based on a non-local charge transfer," J. Comput. Phys., Vol. 213, 117-140, 2006.
74. Walker, D. W. and J. J. Dongarra, "MPI: A standard message passing interface," Supercomputer, Vol. 12, 56-68, 1996.
75. Stone, E., D. Gohara, and G. Shi, "OpenCL: A parallel programming standard for heterogeneous computing systems," Computing in Science and Engineering, Vol. 12, 66-72, 2010.
76. Kahan, W., "Further remarks on reducing truncation errors," Communications of the ACM, Vol. 8, 40-40, 1965.
77. ROOT Data Analysis Framework, C++ program package, developed in CERN, https://root.cern.ch/, .
78. Hwu, W.-M. W., (Editor), GPU Computing Gems, Jade Edition, 1st Ed., Morgan Kaufmann, 2011.