Construction of equilibrated singular basis functions without a priori knowledge of analytical singularity order

In this paper a novel method is presented to construct singular basis functions for solving harmonic and bi-harmonic problems with weak singularities. Such bases are found without any knowledge of the singularity order. The singular bases are constructed by choosing a series as a tensor product of Chebyshev polynomials and trigonometrical functions, in radial and angular directions respectively, and applying weak form of the governing equation. With such features, the singular bases are categorized as the equilibrated basis functions. The constructed singular functions can be utilized as a complementary part to the smooth part of the approximation in the solution of problems with singularities. To demonstrate the efficiency of employing such singular bases, they are used in a boundary node method. Through the solution of some examples, selected from the well-known literature, the capability of the method is shown. It will be demonstrated that the main function and its derivatives are excellently approximated at very close neighborhood of the singular point. The method may especially be found useful for those who research on the eXtended Finite Element Method or similar ideas.