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A new method for evaluating singular integrals in stress analysis of solids by the direct boundary element method. (English) Zbl 0576.65129

Problems in stress analysis of elastic and elasto-plastic solids can be formulated in terms of integral equations defined on the boundary. For an approximate solution the boundary is divided into elements, leading to the boundary element method (BEM), where the integral equations are replaced by a system of linear algebraic equations with matrix elements derived by evaluating (singular) integrals over the elements. In this paper triangle polar co-ordinates are used to reduce the order of singularity of the boundary integrals by one degree and to carry out the integration over mappings of the boundary elements onto plane squares. The method is extended to the cubature of singular integrals over three- dimensional internal cells. The analytical investigations, which are explained in detail, are illustrated by numerical examples. The paper is of interest for engineers using BEM for solving stress problems in solids.
Reviewer: S.Christiansen

MSC:

65R20 Numerical methods for integral equations
65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
74B99 Elastic materials
74C99 Plastic materials, materials of stress-rate and internal-variable type
45E05 Integral equations with kernels of Cauchy type
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[1] Cruse, Int. J. Solids Struct. 5 pp 1259– (1969)
[2] ’A further development of the boundary integral technique for elastostatics’, Ph.D. thesis, Univ. of Southampton (1975).
[3] Kutt, Numer. Math. 24 pp 205– (1975)
[4] , , , ’Isoparametric quadratic boundary elements in 2D plasticity’, in Proc. Int. Conf. on Finite Element Methods (, Eds.), Science Press, Beijing, and Gordon and Breach, New York, 1982, pp. 761-765.
[5] ’Higher-order, special and singular-finite elements’, chap. 4, in Survey of Finite Element Methods ( and , Eds.), A. S. M. E., 1980.
[6] Ying, Int. j. numer. methods eng. 18 pp 959– (1982)
[7] Lachat, Int. j. numer. methods eng. 10 pp 991– (1976)
[8] ’Boundary integral equation method for three dimensional elastostatic problems and a formulation of the problem for large displacement’, in Proc. 14th Jugosl. Kongresracion: Primenjene mechanike C, 113-120, 1978.
[9] ; and , ’Method of boundary integral equations for analysis of three dimensional crack problems’, in Boundary Element Methods, Proc. Third Int. Seminar on B.E.M. (ed.), Springer-Verlag, Berlin, 1982, pp. 183-205.
[10] , , ’An implementation of stress discontinuity in the boundary element method’, in Boundary Element Methods, Proc. Third Int. Seminar on B.E.M. (Ed.), Springer-Verlag, Berlin, 1982, pp. 233-253.
[11] and , ’New developments in elastoplastic analysis’ in Boundary Element Methods, Proc. Third Int. Seminar on B.E.M. (Ed.), Springer-Verlag, Berlin, 1982, pp. 350-370.
[12] and , ’Elasto-plastic boundary element analysis’, in Nonlinear Finite Element Analysis in Structural Mechanics (), and , (Eds.), Springer-Verlag, Berlin, 1981, pp. 403-434.
[13] and , Boundary Element Methods in Engineering Science, McGraw-Hill, New York, 1981. · Zbl 0499.73070
[14] ’Solutions of certain boundary integral equations in potential theory’, Ph.D. thesis, City Univ., London (1971).
[15] and , Mathematical Hand Book for Scientists and Engineers, Appendix E, McGraw-Hill, New York, 1968.
[16] ’Elastostatics’, chap. 4, in Progress in Boundary Element Methods, vol. 1 (Ed.), Pentech Press, London, 1981, pp. 84-167.
[17] and , Theory of Elasticity, 2nd edn, McGraw-Hill, New York, 1951, pp. 356-362. · Zbl 0054.01202
[18] Beer, Int. j. numer. methods eng. 17 pp 43– (1981)
[19] Lachat, Comp. Meth. Appl. Mech. Eng. 10 pp 273– (1976)
[20] and , ’A boundary integral method for the numerical computation of the forces exerted on a sphere in viscous, incompressible flows near a plane wall’, Preprint-Nr. 872, Technische Hochschule Darmstadt, Fachbereich Mathematik, 1985, to appear also in ZAMP.
[21] and , ’3-D BEM and numerical integration’, in Proc. 7th Conference on B.E.M. ( and , Eds.), Springer-Verlag, Berlin, 1985, in press.
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