As hitherto proposed in the technical literature, the boundary element modelling of cracks is best carried out resorting to a hypersingular fundamental solution – in the frame of the so-called dual formulation –, since with the singular fundamental solution alone the ensuing topological issues would not be adequately tackled. A more natural approach might rely on the direct representation of the crack tip singularity, as already proposed in the frame of the hybrid boundary element method – with implementation of generalized Westergaard stress functions. On the other hand, recent mathematical assessments indicate that the conventional boundary element formulation – based on Kelvin’s fundamental solution – is in fact able to precisely represent high stress gradients and deal with extremely convoluted topologies provided only that the numerical integrations be properly resolved. We propose in this work that independently of configuration a cracked structure be geometrically represented as it would appear in laboratory experiments, with crack openings in the range of micrometers. (The nanometer range is actually mathematically feasible in the present formulation but not realistic in terms of continuum mechanics.) Owing to the newly developed numerical integration scheme, machine precision evaluation of all quantities may be achieved and stress results consistently evaluated at interior points arbitrarily close to crack tips. Importantly, no artificial topological issues are introduced, linear algebra conditioning is well kept under control and arbitrarily high convergence of results is always attainable. The present developments apply to two-dimensional problems. Some numerical illustrations show that highly accurate results are obtained for cracks represented with just a few quadratic, generally curved, boundary elements – and a few Gauss-Legendre integration points per element – and that the numerical evaluation of the J-integral turns out to be straightforward (although not computationally cheap) and actually the most reliable means of obtaining stress intensity factors.