The Nearly Singular Integrals On High-order Boundary Elements In 3D BEM

The effective estimation of nearly singular integrals on high-order boundary elements is a difficult task in 3D BEM.For most of the current numerical methods,the geometry of the boundary element is often depicted by using lower-order shape functions when nearly singular integrals need to be calculated.Nevertheless,most engineering processes occur mostly in complex geometrical domains,and obviously,higher order geometry elements are expected to be more accurate,especially for extra-thin structures.Therefore,it needs to be further investigated to accurately evaluate the nearly singular integrals on high-order boundary elements.This paper is devoted to the research on the efficient treatment of nearly singular integrals in 3D BEM.High-order boundary elements,usually of second-order,are employed to approximate the geometric boundary of the structures.Based on this,a novel distance function,which is cconvenient for the implementation of transformation,is constructed to accurately approximate the real distance.Then,an extended form of exponential transformation is developed to remove the near singularities.Numerical tests demonstrate that the present method could enormously raise the computational accuracy of nearly singular integrals.Moreover,it becomes possible to deal with the three-dimensional thin-body problems due to the usage of high-order boundary elements.In present paper,the proposed algorithm is firstly applied to dispose of the boundary layer effect in 3D potential problem and elasticity problem.The physical quantities at inner points which are close to the boundary could be accurately computed by the current method even the distance between the inner point and the boundary equals to 1.0E-9.Secondly,thin-body problems occurring in three-dimensional potential and elastic problems are taken into account.Generally,numerical analysis of the behavior of these structres represents a great challenge to researchers in engineering applications due to the small size of its thickness.In the present work,the unknown physical quantities on the boundary are precisely computed in the first step.Then the inner physical quantities are further studied.Compared with the previous methods,the new algorithm can solve problems whose thickness-to-length ratios are smaller since the nearly integrand has been fully regularized.Several numerical examples demonstrate the high efficiency and stability of present method.A general strategy is proposed for calculation of the nearly singular integrals 3D boundary element analysis.The present algorithm can be applied to any high-order geometric boundary elements and any high-order interpolation elements.Due to the usage of the proposed method,both the three-dimensional boundary layer effect and thin-body problems can be successfully overcome.