Complete Cubic Polynomial Transformation for Regularizing Nearly Hypersingular Integrals

A classical problem in computational mechanics is the quadrature of nearly singular and hypersingular integrals, such as those that arise in the boundary element method (BEM) when the collocation point is near the integration element. Due to the rapid variation of the integrand in these nearly singular and nearly hypersingular cases, standard procedures cannot compute the integrals accurately. To address this, a complete cubic polynomial transformation is developed in this work, significantly outperforming the Telles cubic polynomial, which is traditionally regarded as the reference cubic transformation for regularizing such integrals. In contrast with the Telles polynomial, which includes a free parameter determined through an ad hoc optimization procedure, the new polynomial proposed here is complete and contains no free parameters, as all of them result from a rational method. This enhancement is achieved by regularizing the 1/r2 singularity at the complex pole within isoparametric elements, which leads to a fully defined cubic polynomial transformation whose Jacobian mirrors the isolated singularity and vanishes at the complex pole, effectively neutralizing it. Numerical integration results over canonical straight and curved elements are presented, comparing standard numerical integration, the Telles cubic polynomial, and the new cubic polynomial for near singularities of the kind ln 1/r, 1/r, 1/r2, 1/r3, ultimately demonstrating the superior performance of the proposed method.