Hybrid nearly singular integration for three-dimensional isogeometric boundary element analysis of coatings and other thin structures

The isogeometric boundary element method (IGABEM) has great potential for the simulation of elasticity problems because of its exact geometric representation and good approximation properties. These advantages can be extended to thin structures, including coatings, but the development of an accurate and efficient method to deal with the large number of nearly singular integrals existing in the IGABEM presents a great challenge for very thin sections. In this paper, we propose a new sinh⁺ scheme for weakly, strongly and hyper near-singular integrals arising in 3D IGABEM for thermoelastic problems, based on the sinh transformation method and adaptive integral method. The presented scheme is efficient, since it combines the advantages of both methods: (1) when the thickness δ of coatings/thin structures is moderately small, an accurate and efficient integral result will be obtained by the adaptive integral method; (2) when δ is very small, the nearly singular integrals are computed by the sinh⁺ scheme efficiently. With the introduction of NURBS in IGABEM, truncation errors arising in the Taylor expansion cannot be ignored. Based on the values of these errors, the computed knot spans are further divided into several sub-knot spans and different methods will be used to evaluate the integral over each sub-knot span in the new scheme. In addition, based on the analytical extension of the NURBS surface, an adaptation of the sinh transformation method is proposed which can evaluate the near-singular integrals accurately for cases in which the projection point lies outside of the considered knot span. Several numerical examples are presented to validate the accuracy and efficiency of the 3D IGABEM based on the sinh⁺ scheme in the analysis of thermoelastic problems.