Numerical Evaluation of Weakly Near-Singular Integrals in Time-Domain Integral Equations

A new singularity cancellation transformation is presented for calculating weakly near-singular integrals in time-domain integral equations. On one hand, the singularity cancellation method is a numerical technique for singularity treatments, which is relatively simple and kernel-independent. The integral kernels of time-domain integral equations depend on the temporal basis functions and solution methodologies, which benefit from the generality of cancellation transformations. On the other hand, the singularity cancellation transformations for weakly near-singular integrals are inefficient on the deformed triangular domain, which is referred to as shape-dependence in this paper. The shape-dependence issue is a main problem of singularity cancellation methods. Therefore, the reason of shape-dependence is investigated via the theoretical analysis in this work. Second, a new singularity cancellation transformation is proposed for calculating weakly near-singular integrals in time-domain integral equations, which have fast and consistent convergence rate for both regular and irregular triangles. Some numerical results are given to show the effectiveness of the proposed variable transformation.