位勢邊界元法中的邊界層效應與薄體結構

Abstract

邊界層效應與薄體結構問題的數值分析是邊界元法的難點之一，其實質是近奇異積分的精確計算.現有的處理近奇異積分的多數方法，特別是精確積分法，通常考慮的是線性幾何單元.然而，多數工程問題的幾何區域是十分複雜的，採用高階幾何單元近似顯然能更好地逼近問題的真實邊界，所得結果也將更加精確.但由於高階幾何單元下的雅可比及被積函數形式的複雜性，相應的近奇異積分的精確計算一直是一個非常困難的問題.提出一種新的反插值思想和方法，將被積函數中的規則部分用反插值多項式近似，從而導出計算近奇異積分的精確表達式.數值算例表明，該算法穩定，效率高，在不增加計算量的前提下，極大地改進了近奇異積分計算的精度，成功地解決了邊界層效應與薄體結構問題。In boundary element analysis, when a considered field point is very close to an integral element, the kernels' integration would various levels of near singularity, which can not be computed accurately with the standard Gaussian quadrature. As a result, the numerical results of field variables and their derivatives may become less satisfactory of even out of true. This is so-called "boundary layer effect". Therefore, the accurate evaluation of nearly singular integrals plays an essential role to obtain highly accurate and reliable results by using boundary element method(BEM). For most of the current numeriacla methods, especially for the exact integration method, the geometry of the boundary element is often depicted by using linear shape functions when nearly singular integrals need to be calculated. However, most engineering processes occur mostly in complex geometrucal domains, and obviously, higher order geometry elements are expected to be more accurate to slove such practical problems. Thus, efficient approaches for estimating nearly singular integrals with high order geometry elements are necessary both in theory and application, and need to be further investigated. As is well known, for high order geometry elements, the forms of Jacobian and integrands are all complex irrational functions, and thus for a long time, the exact evaluation of nearly singular integrals is a difficult problem or even impossible implementation. In this paper, a new exact integration method for element integrals with the curvilinear geometru is presented. The present method can greatly improve the accuracy of numerical results of near singular integrals withour increasing other computational efforts. Numerical examples of potential problems with curved elements demonstrate that the present algorithm can effectively handle singular integrals occuring in boundary layer effect and thin body problems in BEM.