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

Abstract

SUMMARYThis 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.

A practical examination of the errors arising in the direct collocation boundary element method for acoustic scattering

“…These include analytical evaluation [41,42], division of the interval about the singularity [43], subtraction of the singularity [42], special quadrature routines [44], and the use of non-linear coordinate transformations [45][46][47][48][49]. Each method has its own particular advantages and disadvantages depending on the type and order of the boundary elements used, and the form of the singular integral.…”

“…Of the multitude of available non-linear coordinate transformations, that proposed by Sato and colleagues is amongst the most efficient [43]. The original Sato transformation of order sZ2 is given by…”

“…For interior singularities, the interval must first be partitioned at the singularity, the transformation applied to both partitions, and the intervals then mapped back to [À1,1] [43]. For an arbitrary interior singularity located at À1oxo1, this yields the following general transformation…”

“…Even for algorithms suitable for interior point singularities, partitioning can often improve the accuracy of the solution for a given number of overall Gaussian operations [43]. Note, that for an endpoint singularity, the Sato transformations for s ¼ 2, 3 are identical to the popular transformations proposed by Telles [45].…”

“…The efficiency of the Sato transformation for weakly singular integrals with a logarithmic singularity and real-valued kernels has been investigated previously [43]. However, the application of coordinate transformations for complex valued kernels does not always yield the expected results.…”