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-ordinat e transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence an d accuracy of the available non-linear polynomial transformations is presen ted for two-dimensional problems. Effectiveness of generalized transformati ons valid for any type and location of singularity has been investigated. I t is found that weakly singular integrals are more efficiently handled with transformations valid for end-point singularities by partitioning the elem ent at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connectio n between the maximum permissible order of polynomial transformations and p recision of computations has also been investigated; cubic transformation i s seen to be the optimum choice for single precision, and quartic or quinti c 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, efficient a nd robust than the singularity subtraction method and the non-linear transf ormation methods. Copyright (C) 2001 John Wiley & Sons, Ltd.