A new algorithm for near–singular integration of 3D Boundary Element Integrals for thin–walled elements

  • Marco Antonio González De León Universidad Simón Bolívar
  • Luiz Carlos Wrobel Brunel University
  • Manuel del Jesús Martinez Universidad Central de Venezuela
Keywords: MEC, placas, carcasas, integración numérica, transformación polinómica, BEM, plates, shells, numerical integration, polynomial transformation

Abstract

The accuracy of the Boundary Element Method (BEM) is strongly dependent on an accurate evaluation of boundary integrals. For thin-walled structures, collocation points and integration elements are often very close, making the kernel of the integrations nearly singular and requiring the use of special numerical integration techniques. In this paper, an effective algorithm is presented for near–singular integration of boundary element integrals applied to three–dimensional thin-walled structures. A combination of Telles’ transformation of variables technique and an adaptive Gaussian quadrature method for regular integrals is used to improve the integration accuracy and to decrease the computation time. The choice of parameters for the technique depends on the relationship between the distance from collocation point to integration element and a reference element length. The proposed integration algorithm is applied to thin plate uniaxial loading and pressurized thin-walled cylindrical shells. The results obtained are in good agreement with theoretical results and the reduction in integration times is significant.

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Author Biographies

Luiz Carlos Wrobel, Brunel University

Professor, Deputy Head of School - Research

School of Engineering and Design

 

Manuel del Jesús Martinez, Universidad Central de Venezuela
Profesor Titular de la Escuela de Ingeniería Mecánica. Director de Postgrado de la Facultad de Ingeniería

References

L. Wrobel, M. Aliabadi. The Boundary Element Method. 1st ed. Ed. John Wiley & Sons. New York, USA. 2002. pp. 491-523.

A. Becker. Boundary Element Method in Engineering: A Complete Course. 1st ed. Ed. McGraw Hill. London, UK. 1992. pp. 1-14.

A. Zehnderand, M. Viz. “Fracture mechanics of thin plates and shells under combined membrane, bending and twisting loads”. Applied Mechanics Reviews. Vol. 58. 2005. pp. 37-48.

C. Providakis, D. Beskos. “Dynamic analysis of plates by boundary elements”. Applied Mechanics Reviews. Vol. 52. 1999. pp. 213-236.

Y. Liu. “Analysis of shell-like structures by the Boundary Element Method based on 3-D elasticity: formulation and verification”. International Journal for Numerical Methods in Engineering. Vol. 41. 1998. pp. 541-558.

J. Lachat, J. Watson. “Effective numerical treatment of Boundary Integral Equations: A formulation for threedimensional elastostatics”. International Journal for Numerical Methods in Engineering. Vol. 10. 1976. pp. 991-1005.

J. Telles. “A self-adaptive co-ordinate transformation for efficient numerical evaluation of general boundary element integrals”. International Journal for Numerical Methods in Engineering. Vol. 24. 1987. pp. 959-973.

W. Ye. “A new transformation technique for evaluating nearly singular integrals”. Computational Mechanics. Vol. 42. 2008. pp. 457-466.

Y. Gu, W. Chen, C. Zhang. “The sinh transformation for evaluating nearly singular boundary element integrals over high-order geometry elements”. Engineering Analysis with Boundary Elements. Vol. 37. 2013. pp. 301-308.

P. Johnston, B. Johnston, D. Elliott. “Using the iterated sinh transformation to evaluate two dimensional nearly singular boundary element integrals”. Engineering Analysis with Boundary Elements. Vol. 37. 2013. pp. 708-718.

B. Johnston, P. Johnston, D. Elliott. “A new method for the numerical evaluation of nearly singular integrals on triangular elements in the 3D boundary element method”. Journal of Computational and Applied Mathematics. Vol. 245. 2013. pp. 148-161.

J. Rong, L. Wen, J. Xiao. “Efficiency improvement of the polar coordinate transformation for evaluating BEM singular integrals on curved elements”. Engineering Analysis with Boundary Elements. Vol. 38. 2014. pp. 83-93.

Y. Zhang, X. Li, V. Sladek, J. Sladek, X. Gao. “Computation of nearly singular integrals in 3D BEM”. Engineering Analysis with Boundary Elements. Vol. 48. 2014. pp. 32-42.

Y. Liu, D. Zhang, F. Rizzo. Nearly singular and hypersingular integrals in the Boundary Element Method. Boundary Elements XV, Proceedings of the 15th Int. Conf. on Boundary Elements. Massachusetts, USA. 1993. pp. 453-468.

S. Mukherjee, M. Chati, X. Shi. “Evaluation of nearly singular integrals in boundary element contour and node methods for three-dimensional linear elasticity”. International Journal of Solids and Structures. Vol. 37. 2000. pp. 7633-7654.

Z. Niu, W. Wendland, X. Wang, H. Zhou. “A semianalytical algorithm for the evaluation of the nearly singular integrals in three-dimensional boundary element methods”. Computer Methods in Applied Mechanics and Engineering. Vol. 194. 2005. pp. 1057- 1074.

H. Hosseinzadeh, M. Dehghan. “A simple and accurate scheme based on complex space C to calculate boundary integrals of 2D boundary elements method”. Computers & Mathematics with Applications. Vol. 68. 2014. pp. 531-542.

H. Ma, N. Kamiya. “A general algorithm for the numerical evaluation of nearly singular boundary integrals of various orders for two- and threedimensional elasticity”. Computational Mechanics. Vol. 29. 2002. pp. 277-288.

G. Xie, J. Zhang, X. Qin, G. Li. “New variable transformations for evaluating nearly singular integrals in 2D boundary element method”. Engineering Analysis with Boundary Elements. Vol. 35. 2011. pp. 811-817.

G. Xie, F. Zhou, J. Zhang, X. Zheng, C. Huang. “New variable transformations for evaluating nearly singular integrals in 3D boundary element method”. Engineering Analysis with Boundary Elements. Vol. 37. 2013. pp. 1169-1178

Z. Niu, C. Cheng, H. Zhou, Z. Hu. “Analytic formulations for calculating nearly singular integrals in two-dimensional BEM”. Engineering Analysis with Boundary Elements. Vol. 31. 2007. pp. 949-964.

F. Araújo, K. Silva, J. Telles. “Generic domain decomposition and iterative solvers for 3D BEM problems”. International Journal for Numerical Methods in Engineering. Vol. 68. 2006. pp. 448-472.

H. Li, G. Han, H. Mang. “A new method for evaluating for evaluating singular integrals in stress analysis of solids by the direct boundary element method”. International Journal for Numerical Methods in Engineering. Vol. 21. 1985. pp. 2071–2098.

R. Rigby. Boundary Element Analysis of Cracks in Aerospace Structures. PhD Thesis, University of Portsmouth. Portsmouth, UK. 1995. pp. 25-32

X. Qin, J. Zhang, G. Xie, F. Zhou, G. Li. “A general algorithm for the numerical evaluation of nearly singular integrals on 3D boundary element”. Journal of Computational and Applied Mathematics. Vol. 235. 2011. pp. 4174-4186.

S. Mukherjee. “Integral equation formulation for thin shells—revisited”. Engineering Analysis with Boundary Elements. Vol. 31. 2007. pp. 539-546.

F. Araújo, K. Silva, J. Telles. “Application of a generic domain-decomposition strategy to solve shell-like problems through 3D BE models”. Communications in Numerical Methods in Engineering. Vol. 23. 2007. pp. 771-785.

A. Frangi, M. Guiggiani. “Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals”. International Journal for Numerical Methods in Engineering. Vol. 46. 1999. pp. 1845-1863.

M. Guiggiani. “The evaluation of Cauchy principal value integrals in the boundary element method – A review”. Mathematical and Computer Modelling. Vol. 15. 1991. pp. 175-184.

D. Beskos. “Boundary element methods in dynamic analysis: Part II (1986-1996)”. Applied Mechanics Reviews. Vol. 50. 1997. pp. 149-197.

N. Dowling. Mechanical Behavior of Materials. Engineering Methods for Deformation, Fracture, and Fatigue. 3rd ed. Ed. Prentice Hall. New Jersey, USA. 2007. pp. 785-786.

Published
2015-02-18
How to Cite
González De León M. A., Wrobel L. C., & Martinez M. del J. (2015). A new algorithm for near–singular integration of 3D Boundary Element Integrals for thin–walled elements. Revista Facultad De Ingeniería Universidad De Antioquia, (74), 96-107. Retrieved from https://revistas.udea.edu.co/index.php/ingenieria/article/view/17960