Abstract:
In boundary element analyses, when a considered fieldpoint is very close to an integral element, the kernels' integration wouldexist various levels of near singularity, which can not be computedaccurately with the standard Gaussian quadrature. As a result, the numericalresults of field variables and their derivatives may become lesssatisfactory or even out of true. This is so-called ``boundary layereffect''. Therefore, the accurate evaluation of nearly singular integralsplays an essential role to obtain highly accurate and reliable results byusing boundary element method (BEM). For most of the current numericalmethods, especially for the exact integration method, the geometry of theboundary element is often depicted by using linear shape functions whennearly singular integrals need to be calculated. However, most engineeringprocesses occur mostly in complex geometrical domains, and obviously, higherorder geometry elements are expected to be more accurate to solve suchpractical problems. Thus, efficient approaches for estimating nearlysingular integrals with high order geometry elements are necessary both intheory and application, and need to be further investigated. As is wellknown, for high order geometry elements, the forms of Jacobian andintegrands are all complex irrational functions, and thus for a long time,the exact evaluation of nearly singular integrals is a difficult problem oreven impossible implementation. In this paper, a new exact integrationmethod for element integrals with the curvilinear geometry is presented. Thepresent method can greatly improve the accuracy of numerical results ofnearly singular integrals without increasing other computational efforts.Numerical examples of potential problems with curved elements demonstratethat the present algorithm can effectively handle nearly singular integralsoccurring in boundary layer effect and thin body problems in BEM.
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