On non-linear transformations for accurate numerical evaluation of weakly singular boundary integrals

This paper presents a study of the performance of the non-linear co-ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non-linear polynomial transformations is presented for two-dimensional problems. E ectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more e ciently handled with transformations valid for end-point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non-linear transformation has been proposed. This composite approach is found to be more accurate, e cient and robust than the singularity subtraction method and the non-linear transformation methods