Quadrature Rules for Functions with a Mid-Point Logarithmic Singularity in the Boundary Element Method Based on the <i>x = t<sup>p</sup></i> Substitution — Oak Academic Publishing
Research ArticleOpen AccessGoogle Scholar indexed
Quadrature Rules for Functions with a Mid-Point Logarithmic Singularity in the Boundary Element Method Based on the <i>x = t<sup>p</sup></i> Substitution
School of Engineering, University of Central Lancashire, Preston, UK
,
School of Engineering, University of Central Lancashire, Preston, UK
,
Hellenic Ministry of National Defence, Athens, Greece
1 School of Engineering, University of Central Lancashire, Preston, UK
2 School of Engineering, University of Central Lancashire, Preston, UK
3 Hellenic Ministry of National Defence, Athens, Greece
Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1, 1], with a logarithmic singularity at the centre. The substitution x = t p , where p (≥ 3) is an odd integer is given particular attention, as this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p . It is shown that a high value for p typically gives more accurate results.
KeywordsBoundary Element MethodSingular IntegralNumerical Integration
Ang, W.-T. (2007) A Beginner’s Course in Boundary Element Methods. Universal Publishers, Boca Raton.
Wrobel, L.C. (2002) The Boundary Element Method—Volume 1—Applications in Thermo-Fluids and Acoustics. John Wiley and Sons, Hoboken. https://doi.org/10.1115/1.1553431
Boundary Element Method. http://www.boundary-element-method.com
Salvadori, A. (2002) Analytical Integrations in 2D BEM Elasticity. International Journal for Numerical Methods in Engineering, 53, 1695-1719. https://doi.org/10.1002/nme.359
Johnston, P. and Elliot, D. (2005) A Sinh Transformation for Evaluating Nearly Singular Boundary Element Integrals. International Journal for Numerical Methods in Engineering, 62, 564-578. https://doi.org/10.1002/nme.1208
Sikora, J., Polakowski, K. and Panczyk, B. (2017) Numerical Calculation of Singular Integrals for Different Formulations of Boundary Element. Przeglad Elektrotechniczny, 93, 181-185.
Salvadori, A. (2010) Analytical Integrations in 3D BEM for Elliptic Problems: Evaluation and Implementation. International Journal for Numerical Methods in Engineering, 84, 505-542. https://doi.org/10.1002/nme.2906
Kirkup, S.M. and Henwood, D.J. (1986) Practical Numerical Methods for the Integration of Functions of One Variable with an End Point Singularity. Report M.11, Department of Mathematics, Statistics and Operational Research, Brighton Polytechnic, Brighton.
Kirkup, S.M. (1989) Solution of Exterior Acoustic Problems by the Boundary Element Method. PhD Thesis, Brighton Polytechnic, Brighton.
Kirkup, S.M. and Henwood, D.J. (1994) An Empirical Analysis of the Boundary Element Method Applied to Laplace’s Equation. Applied Mathematical Modelling, 18, 32-38. https://doi.org/10.1016/0307-904X(94)90180-5
Kirkup, S.M. (1998) Fortran Codes for Computing the Discrete Helmholtz Integral Operators. Advances in Computational Mathematics, 9, 391-409. https://doi.org/10.1023/A:1018953910353
Kirkup, S.M. (2007) The Boundary Element Method in Acoustics. Integrated Sound Software, Hebden Bridge.
Kirkup, S.M. and Yazdani, J. (2008) A Gentle Introduction to the Boundary Element Method in Matlab/Freemat. WSEAS MAMECTIS Conference, Corfu, 46-52.
Telles, J.C.F. (1987) A Self-Adaptive Coordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integrals. International Journal for Numerical Methods in Engineering, 24, 959-973. https://doi.org/10.1002/nme.1620240509
Papazafeiropoulos, G. (2019) Vectorized Numerical Integration Matlab Version 1.2. https://www.mathworks.com/matlabcentral/fileexchange/48931-vectorized-numerical-integration-matlab
Amini, S. and Kirkup, S.M. (1995) Solution of the Helmholtz Equation in the Exterior Domain by Elementary Boundary Integral Methods. Journal of Computational Physics, 118, 208-221. https://doi.org/10.1006/jcph.1995.1093
Burton, A.J. and Miller, G.F. (1971) The Application of Integral Equation Methods to the Numerical Solution of Some Exterior Boundary Value Problems. Proceedings of the Royal Society of London. Series A, 323, 201-210. https://doi.org/10.1098/rspa.1971.0097
Chen, J.T. and Kong, H.-K. (1999) Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series. Applied Mechanics Reviews, 52, 17-33. https://doi.org/10.1115/1.3098922
Kirkup, S.M. (1994) The Boundary and Shell Element Method. Applied Mathematical Modelling, 18, 418-422. https://doi.org/10.1016/0307-904X(94)90302-6
Kirkup, S.M. (1997) Solution of Discontinuous Interior Helmholtz Problems by the Boundary and Shell Element Method. Computer Methods in Applied Mechanics and Engineering, 140, 393-404. https://doi.org/10.1016/S0045-7825(96)01117-6
Kirkup, S.M. (2007) DC Capacitor Simulation by the Boundary Element Method. Communication in Numerical Methods in Engineering, 23, 855-869. https://doi.org/10.1002/cnm.929
Rabinowitz, P. (1987) Numerical Integration in the Presence of an Interior Singularity. Journal of Computational and Applied Mathematics, 17, 31-41. https://doi.org/10.1016/0377-0427(87)90036-7
Davis, P.J. and Rabinowitz, P. (1965) Ignoring the Singularity in Approximate Integration. Journal of the Society for Industrial and Applied Mathematics: Series B, Numerical Analysis, 2, 367-383. https://doi.org/10.1137/0702029
El-Tom, M.E.A. (1971) On Ignoring the Singularity in Approximate Integration. SIAM Journal on Numerical Analysis, 8, 412-424. https://doi.org/10.1137/0708039
Anderson, D.G. (1965) Gaussian Quadrature Formulae for ∫ 0 1 -Inxf(x)dx. Mathematics of Computation, 19, 477-481. https://doi.org/10.2307/2003682
Anselone, P.M. (1981) Singularity Subtraction in the Numerical Solution of Integral Equations. Journal of the Australian Mathematical Society (Series B), 22, 408-418. https://doi.org/10.1017/S0334270000002757
Goodwin, E.T. (1949) The Evaluation of Integrals of the Form ∫_(-∞)^∞?f(x)e -x 2 dx. Proceedings of the Cambridge Philosophical Society, 45, 241-245. https://doi.org/10.1017/S0305004100024786
Takahasi, M. and Mori, M. (1973) Quadrature Formulas Obtained by Variable Transformation. Numerische Mathematik, 21, 206-219. https://doi.org/10.1007/BF01436624
Amini, S. (1986) Efficient Quadrature Rules with a Priori Error Estimates for Integrands with End Point Singularities. BIT, 26, 200-208. https://doi.org/10.1007/BF01933746
Keisan Online Calculator, Nodes and Weights of Gaussian Quadrature. http://keisan.casio.com/exec/system/1329114617
Sikora, J., Polakowski, K. and Pańczyk, B. (2017) Improper Integrals Calculations for Fourier Boundary Element Method. The Applied Computational Electromagnetics Society, 32, 761-768.
Sikora, J., Pańczyk, B. and Polakowski, K. (2017) Numerical Calculation of Singular Integrals for Different Formulations of Boundary Element. Przegląd Elektrotechniczny, 1, 181-185. https://doi.org/10.15199/48.2017.11.37
Gong, J.Y., et al. (2017) Numerical Quadrature for Singular and Near-Singular Integrals of Boundary Element Method and Its Applications in Large-Scale Acoustic Problems. Chinese Journal of Acoustics, 36, 289-301.