A Method for Numerical Evaluation of Singular Integrals in Curved Hexahedra and With High-Order Source Functions

Abstract A method is developed for evaluating the 3-D Biot-Savart singular integral for the velocity field induced by arbitrarily high-order (discontinuous) vorticity in arbitrarily high-order curved hexahedral elements. The proposed method uses Duffy’s coordinate transformation and singularity removal strategy, which, through a set of transformations, accommodates accurate evaluation of the transformed volume integrals using standard adaptive cubature techniques. In this paper, the new method is formulated in detail, followed by a series of benchmark tests demonstrating the convergence properties of the singular volume integral as a function of the discretization order of the vorticity (source) field.