The Regularization Of Nearly Singular Integrals In The 3D BEM And Its Application To Thin-body And Coating Structures

Due to the merits of high accuracy and reducing the number of unknowns by one order,the boundary element method(BEM)has become an appealing area of research and is widely used in engineering problems.However,the existence of nearly singular integral limits its application in a lot of aspects,such as the 3D thin-body and coating structure problems.Although many methods of evaluating two-dimensional(2D)nearly singular integrals have been developed,most of the techniques are used to the lower-order geometry elements,e.g.usual linear or planar elements.In recent years,the computation of the nearly singular integrals over surface elements has achieved great progress,but only the single element integrals are tested.Generally speaking,most of the engineering processes occur in complex geometrical domains,such as thin body and coating structures whose thickness are in the nano-or micro-meters.However,the numerical analysis of these problems by BEM is always a great challenge.In recent years,scholars found that the bottleneck of the problems is the evaluation of nearly singular integral on higher-order elements.In this paper,we present an efficient strategy for numerical evaluation of 2D nearly singular integrals that arise in the solution of 3D BEM using eight-node second-order quadrilateral surface elements.The main novel featur es of the proposed method lie in:(1)Developing an ‘exact distance formula.Such distance formula is easy to combine with the sinh transformation.Next,it is able to accurately approximate the actual distance r,from the source point to a generic point of the element.So promising results for numerical evaluation of 2D nearly singular integrals can be expected;(2)Ingenious combination of the aforementioned distance formula with the sinh transformation.The sinh transformation can smooth out the rapid variation of the developed distance formula on integration interval.The presented method is applied to 3D thin-body and coating structure problems and satisfactory results are obtained.The numerical results show that the proposed scheme can deal with the most general surface elements(the eight-node second-order quadrilateral geometry elements).And the proposed scheme make it possible to solve thin-body,coating structure problems and the boundary layer effect.In this paper,the thin-body and coating structures problems are solved,which also extends the application scope of the BEM.