Numerical evaluation of nearly hyper-singular integrals in the boundary element analysis

•Reviewing the numerical techniques used currently to calculate the nearly singular integrals.•Some incorrect algorithms published before are discussed.•Proposing a new method that is suitable for a wide range of nearly singular integrals.•Particular focus is on integrals with near hyper-singularities and beyond.

Accurate evaluation of nearly singular integrals plays an important role in the overall accuracy of the boundary element analysis. The paper presented here reviews some numerical techniques used currently to calculate such integrals in the boundary element method (BEM). Some incorrect algorithms published before are discussed and a new numerical technique to calculate nearly singular integrals with hyper-singularities is developed. The accuracy and efficiency of the method are demonstrated through several examples that are commonly encountered in the applications of the BEM. Comparison of this method with some of the existing methods is also presented. It is shown that, for the evaluation of nearly hyper-singular integrals, several orders of magnitude improvement in relative error can be obtained using the current method compared to most of existing methods.