A general algorithm for the numerical evaluation of nearly singular boundary integrals in the equivalent non-singular bies with indirect unknowns

Journal of the Chinese Institute of Engineers, Transactions of the Chinese Institute of Engineers,Series A/Chung-kuo Kung Ch'eng Hsuch K'an

Volumn 31, Issue 3, 2008, Pages 437-447

  Zhang, Yaoming ^a   Sun, Cuilian ^a  

^a SHANDONG UNIVERSITY OF TECHNOLOGY   (China)

Author keywords

Boundary layer effect; Nearly singular integrals; Potential problem; Transformation

Indexed keywords

BOUNDARY LAYERS;

BOUNDARY ELEMENT ANALYSIS; BOUNDARY LAYER EFFECTS; COMPUTATIONAL EFFORT; NEARLY SINGULAR INTEGRAL; NUMERICAL RESULTS; POTENTIAL PROBLEMS; SINGULAR BOUNDARY; TRANSFORMATION;

BOUNDARY ELEMENT METHOD;

EID: 44949206435     PISSN: 02533839     EISSN: 21587299     Source Type: Journal    
DOI: 10.1080/02533839.2008.9671398     Document Type: Article

References (52)

1. Brebbia, C. A., Tells, J. C. F., and Wrobel, L. C., 1984. Boundary Element Technique, Berlin, Germany:Springer-Verlag.
2. Cerrolaza, M., and Alarcon, E., 1989. “A Bi-Cubic Transformation for the Numerical Evaluation of the Cauchy Principal Value Integrals in Boundary Methods,”. International Journal for Numerical Methods in Engineering, 28 (5):987–999.
3. Chen, H. B., Lu, P., Huang, M. G., and Williams, F. W., 1998. “An Effective Method for Finding Values on and Near Boundaries in the Elastic BEM,”. Computers & Structures, 69 (4):421–431.
4. Chen, H. B., Lu, P., and Schnack, E., 2001. “Regularized Algorithms for the Calculation of Values on and Near Boundaries in 2D Elastic BEM,”. Engineering Analysis with Bondary Elements, 25 (10):851–876.
5. Chen, J. T., and Hong, H. K., 1994. “Dual Boundary Integral Equations at a Corner Using Contour Approach around Singularity,”. Advances in Engineering Software, 21 (3):169–178.
6. Chen, J. T., and Hong, H. K., 1999. “Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series,”. Applied Mechanics Reviews, 52 (1):17–33.
7. Chen, J. T., Lin, T. W., Chen, K. H., and Chyuan, S. W., 2004. “True and Spurious Eigensolutions for the Problems with the Mixed-Type Boundary Conditions Using BEMs,”. Finite Element in Analysis and Design, 40 (11):1521–1549.
8. Chen, J. T., Lin, T. W., Chen, I. L., and Lee, Y. J., 2005. “Fictitious Frequency for the Exterior Helmholtz Equation Subject to the Mixed-type Boundary Condition Using BEMs,”. Mechanics Research Communications, 32 (1):75–92.
9. Chen, J. T., Shen, W. C., and Wu, A. C., 2006a. “Nullfield Integral Equations for Stress Field around Circular Holes under Antiplane Shear,”. Engineering Analysis with Boundary Elements, 30 (3):205–217.
10. Chen, J. T., and Wu, A. C., 2006b. “Null-Field Approach for Piezoelectricity Problems with Arbitrary Circular Inclusions,”. Engineering Analysis with Boundary Elements, 30 (11):971–993.
11. Chen, J. T., and Wu, A.C., 2007a. “Null-Field Integral Equation Approach for Multi-Inclusion Problem under Antiplane Shear,”. ASME, Journal of Applied Mechanics, 74 (3):469–487.
12. Chen, J. T., and Shen, W. C., 2007b. “Degenerate Scale for Multiply Connected Laplace Problems,”. Mechanics Research Communications, 34 (1):69–77.
13. Cruse, T.A., 1974. “An Improved Boundary Integral Equation Method for Three Dimensional Elastic Stress Analysis,”. Computers & Structures, 4 (4):741–754.
14. Cruse, T.A., and Aithal, R., 1993. “Non-Singular Boundary Integral Equation Implementations,”. International Journal for Numerical Methods in Engineering, 36 (2):237–254.
15. Davey, K. T. M., 2004. “Alonso Rasgado. Integration over Simplexes for Accurate Domain and Boundary Integral Evaluation in Boundary Element Methods,”. Computers & Structures, 82 (2):193–211.
16. Granados, J. J., and Gallego, G., 2001. “Regularization of Nearly Hypersingular Integrals in the Boundary Element Method,”. Engineering Analysis with Boundary Elements, 25 (3):165–184.
17. Huang, Q., and Cruse, T. A., 1993. “Some Notes on Singular Integral Techniques in Boundary Element Analysis,”. International Journal for Numerical Methods in Engineering, 36 (15):2643–2659.
18. Johnston, P. R., 1999. “Application of Sigmoidal Transformations to Weakly Singular and Near Singular Boundary Element Integrals,”. International Journal for Numerical Methoeds in Engineering, 45 (10):1333–1348.
19. Johnston, P. R., 2000. “Semi-Sigmoidal Transformations for Evaluating Weakly Singular Boundary Element Integrals,”. International Journal for Numerical Methoeds in Engineering, 47 (10):1709–1730.
20. Jun, L., Beer, G., and Meek, J. L., 1985. “Efficient Evaluation of Integrals of Order Using Gauss Quadrature,”. Engineering Analysis, 2 (3):118–123.
21. Kisu, H., and Kawahara, T., 1988. “Boundary Element Analysis System Based on a Formulation with Relative Quantity,”. In Boundary Element X, Edited by:Brebbia, C.A., Vol. 1, 111–121. Brebbia:Springer-Verlag.
22. Lachat, J. C., and Watson, J. O. 1976. “Effective Numerical Treatment of Boundary Integral Equation, a Formulation for Three-Dimensional Elastostatics,”. International Journal for Numerical Methods in Engineering, 10 (5):991–1005.
23. Liu, Y. J., 1998. “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, 41 (3):541–558.
24. Liu, Y. J., 2000. “On the Simple Solution and Nonsingular Nature of the BIE/BEM-a Review and Some New Results,”. Engineering Analysis with Boundary Elements, 24 (10):789–795.
25. Luo, J. F., Liu, Y. J., and Berger, E. J., 1998. “Analysis of Two-Dimensional Thin Structures (from Micro- to Nano-Scales) Using the Boundary Element Method,”. Computational Mechanics, 22 (5):404–412.
26. Lutz, E. L., 1992. “Exact Gaussian Quadrature Methods for Near-Singular Integrals in the Boundary Element Nethod,”. Engineering Analysis with Boundary Elements, 9 (3):233–245.
27. Lutz, E. L., 1993. “A Mapping Method for Numerical Evaluation of Two-Dimensional Integrals with Singularity,”. Computational Mechanics, 12 (1–2):19–26.
28. Ma, H., and Kamiya, N., 2001. “A General Algorithm for Accurate Computation of Field Variables and its Derivatives Near Boundary in BEM,”. Engineering Analysis with Boundary Elements, 25 (10):843–849.
29. Ma, H., and Kamiya, N., 2002a. “Distance Transformation for the Numerical Evaluation of Near Singular Boundary Integrals with Various Kernels in Boundary Element Method,”. Engineering Analysis with Boundary Elements, 26 (4):329–339.
30. Ma, H., and Kamiya, N., 2002b. “A General Algorithm for the Numerical Evaluation of Near Singular Boundary Integrals of Various Orders for Two- and Three-Dimensional Elasticity,”. Computational Mechanics, 29 (4–5):277–88.
31. Ma, L.F., and Korsunsky, A. M., 2004. “A Note on the Gauss-Jacobi Quadrature Formulae for Singular Integral Equations of the Second Kind,”. International Journal of Fracture, 126:339–405.
32. Mukerjee, S., Chati, M. K., and Shi, X. L., 2000. “Evaluation of Nearly Singular Integrals in Boundary Element Contour and Node Methods for Three-Dimensional Linear Elasticity,”. International Journal of Solids and Structures, 37 (51):7633–7654.
33. Niu, Z. R., Wang, X. X., and Zhou, H. L., 2001. “A general Algorithm for Calculating the Quantities at Interior Points Close to the Boundary by the BEM,”. Chinese Journal of Theoretical and Applied Mechanics, 33 (2):275–283. (in Chinese)
34. Niu, Z. R., and Zhou, H. L., 2004. “The Natural Boundary Integral Equation in Potential Problems and Regularization of the Hypersingular Integral,”. Computer & Structure, 82 (2–3):315–323.
35. Niu, Z. R., Wendl, W. L., Wang, X. X., and Zhou, H. L., 2005. “A Semi-Analytical Algorithm for the Evaluation of the Nearly Singular Integrals in Three- Dimensional Boundary Element Methods,”. Computer Methods in Applied Mechanics and Engineering, 194 (9–11):1057–1074.
36. Padhi, G. S., Shenoi, R. A., Moy, S. S. J., and McCarthy, M. A., 2001. “Analytical Integration of Kernel Shape Function Product Integrals in the Boundary Element Method,”. Computer & Structure, 79 (14):1325–1333.
37. Schulz, H., Schwab, C., and Wendl, W.L., 1998. “The Computation of Potentials Near and on the Boundary by an Extraction Technique for Boundary Element Methods,”. Computer Methods in Applied Mechanics and Engineering, 157 (3–4):225–38.
38. Shiah, Y. C., and Shih, Y. X., 2006. “Anisotropic Heat Conduction Across an Interface Crack/Defect Filled with a Thin Interstitial Medium,”. Engineering Analysis with Boundary Element, 30 (5):325–337.
39. Shiah, Y. C., and Shih, Y. S., 2007. “Regularization of Nearly Singular Integrals in the Boundary Element Analysis for Interior Anisotropic Thermal field Near the Boundary,”. Engineering Analysis with Boundary Element, 30 (2):219–230.
40. Sladek, V., Sladek, J., and Tanaka, M., 1993. “Regularization of Hypersingular and Nearly Singular Integrals in the Potential Theory and Elasticity,”. International Journal for Numerical Methods in Engineering, 36 (10):1609–1628.
41. Sladek, V., Sladek, J., and Tanaka, M., 2000. “Optimal Transformations of the Integration Variables in Computation of Singular Integrals in BEM,”. International Journal for Numerical Methods in Engineering, 47 (7):1263–1283.
42. Sun, H. C., 1999. “Nonsingular Boundary Element Method,”, Dalian:Dalian University of Technology Press. (in Chinese)
43. Tanaka, M., and Matsumoto, T., 1991. “Nakamura M. Boundary Element Method,”, Tokyo:Baifukan Press. (in Japanese)
44. Telles, JCF. 1987. “A Self-Adaptive Coordinate Transformation for Efficient Numerical Evaluation of General Boundary Element Integral,”. International Journal for Numerical Methods in Engineering, 24:959–973.
45. Wang, J. H., and Tsay, T. K., 2005. “Analytical Evaluation and Application of the Singularities in Boundary Element Method,”. Engineering Analysis with Boundary Elements, 29 (3):241–256.
46. Wang, Y. C., Li, H. Q., Chen, H. B., and Wu, Y., 1994. “Calculating the Stresses and Displacements at Arbitrary Points with the Particular Solution Field Method,”. Acta Mechanica Sinica, 26 (2):222–231.
47. Zhang, X. S., and Zhang, X. X., 2004. “Exact Integrations of Two-Dimensional High-Order Discontinuous Boundary Element of Elastostatics Problems,”. Engineering Analysis with Boundary Elements, 28 (7):725–732.
48. Zhang, Y. M., and Sun, H. C., 2000. “Theoretic Analysis on Virtual Boundary Element,”. Chinese Journal of Comptutational Mechanics, 17 (1):56–62. (in Chinese)
49. Zhang, Y. M., and Sun, H. C., 2001. “Analytical Treatment of Boundary Integrals in Direct Boundary Element Analysis of Plane Potential Elasticity Problems,”. Applied Mathematics and Mechanics, 6:664–673. (in Chinese)
50. Zhang, Y. M., and Wen, W. D., 2004. “A Kind of New Nonsingular Boundary Integral Equations for Elastic Plane Problems,”. Chinese Journal of Theoretical and Applied Mechanics, 36 (3):311–321. (in Chinese)
51. Zhang, Y. M., and Lu, H. X., 2006. “Novel Regularized Boundary Integral Equations for Potential Plane Problems,”. Applied Mathematics and Mechanics, 27 (9):1017–1022. (in Chinese)
52. Zhou, H. L., Wang, X. X., and Niu, Z. R., 2003. “Completely Analytical Algorithm of Nearly Singular Integrals in the Boundary Element Method of Potential Problems,”. Journal of University of Science and Technology of China, 33 (4):431–437.