The aim of this presentation is to investigate the largest accuracy that can be obtained with the ordinary 2D boundary integral equations using advanced algorithms for the integration and modifications in the representation of the boundary. The presence of mathematical singularities demands an accurate evaluation of the involved integrals. Analytical and numerical integration schemes are used, including transformations of the domain and extended Gauss quadrature methods. Efficient parametric boundary elements (modified Overhauser) are used and compared with various kinds of boundary elements, some of which have also C-continuity (quadratic spline). All this effort is applied to solve two-dimensional potential problems. Especially the problem of heat transfer through a thick hollow cylinder with external and internal boundaries maintained at constant temperatures. The conclusion of the work is that no effort in integration and in boundary modelling is in vain as it pays off in accuracy obtained. Its main direct applications are: contact problems, fracture mechanics, sensitivity analysis, shape optimisation...
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