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An integration procedure for curved boundary elements. (English) Zbl 0683.73048

Summary: In the directed boundary element method (DBEM) of analysis, much of the numerical computation involves line integrations along a boundary. The integration is usually performed by quadrature, and in recent previous papers by the authors [e.g.: Int. J. Numer. Methods Eng. 28, No.1, 75-93 (1989; Zbl 0675.73049)] it has been shown that integration can in general be carried out analytically, with the results of the integration presented as algebraic functions, and programmed accordingly. These results have, to this time, been presented for straight boundaries, with application to thin-plate bending analysis only.
For an arbitrary shaped curved element, the derivation of closed-form expressions for boundary integrations is not possible. A semi-analytical- numerical procedure, based on representing the boundary as piecewise straight is developed in this paper. In this procedure, the boundary shape is considered independent of the approximation of the boundary variables, and in the sense that each ‘quadratic’ boundary element is considered geometrically to be made up of several piecewise cubic functions, the formulation might be termed ‘super-parametric’. Generally, a curved boundary does not necessarily imply a need for higher-order approximation of boundary variables, but does necessitate that the more complicated shape be accurately modelled.
The integration procedure presented in this paper can be applid in other two-dimensional problems.

MSC:

74S30 Other numerical methods in solid mechanics (MSC2010)
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References:

[1] Abdel-Akher, Int. j. numer. methods. eng. 28 pp 75– (1989)
[2] Abdel-Akher, Int. j. numer. methods. eng. 28 pp 1389– (1989)
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[4] ’On the application of the boundary element method to plate bending analysis’, Ph.D. thesis, Carleton University, Ottawa, Ontario, 1988.
[5] and , ’Thin elastic plates in bending’, in Developments in Boundary Element Method - 4, and (eds.), Applied Science Publishers Ltd., 91-119 (1986). · Zbl 0586.73138
[6] Abdel-Akher, Commun. Appl. Numer. Methods 5 pp 23– (1989)
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