A note on transformation methods for evaluating nearly singular integrals in BEM
The accurate evaluation of nearly singular integrals is one of the major concerned problems in boundary element analysis, and the importance of this problem is often considered next to the singular boundary integrals. The transformation method, as an important regularized algorithm to solve such problem in BEM, has received intensive attention in the past decades. The basic ideas of the existing transformations can be generalized on the whole into three categories: one is removing the nearly zero factor in the denominator of the kernels using another zero factor (usually generated by Jacobian); another one is converting the nearly zero factor in the denominator to be part of the numerator, which benefits form the idea of the reciprocal transformation for the regularization of weakly singular integrals; the last one is based on the idea of diminishing the difference of the orders of magnitude or the scale of change of operational factors. Numerical tests show that the transformations based on the former idea are effective for the calculation of nearly weak singular integrals but not satisfactory for nearly strong singular or nearly hyper-singular integrals. The second kind of transformations can convert nearly singular kernels into normal kernels, but the original regular parts behave nearly singularly after the transformations, so they are suitable only for a case when the regular part of the integrand is constant. The last kind of transformations is effective for nearly weak singular and nearly strong singular integrals. For nearly hyper-singular integrals, this kind of transformations can be used to obtain acceptable results from the point of view of engineering practice. However, in the sense of mathematics, this kind of transformations is not perfectly adequate to produce the accurate results. Thus, the lack of a general transformation for evaluating various kinds of nearly singular integrals is still a problem to be solved in BEM. The purpose of this note is to discuss the validity of different kinds of transformations and present a general strategy, termed “double transformation, to improve the applicability for various kinds of nearly singular integrals. The proposed strategy can remove the near singularity efficiently and improve the accuracy of numerical results of nearly singular integrals greatly without increasing other computational efforts. Nearly weak, nearly strong as well as nearly hyper-singular integrals can all be calculated accurately by using the presented transformation. Numerical examples of potential problems demonstrate that the present algorithm can effectively handle nearly singular integrals occurring in boundary layer effect and thin body problems in BEM.
Yaoming Zhang Yan Gu Zhaoyan Liu
Shandong University, China
国际会议
南京
英文
51-61
2009-10-12（万方平台首次上网日期，不代表论文的发表时间）