基于多边形网格的变系数问题边界单元法

Polygonal mesh-based boundary element method for variable coefficient problems

提出一种用多边形网格计算二维变系数问题域积分的新型边界单元法。首先,构造了由任意多边组成的多边形网格形函数,用于几何与物理量的插值;其次,用径向积分法将多边形域积分转换成沿多边形周边的线积分,有效解决了各类非规则多边形网格的单元积分难题;最后,三个有关功能梯度材料与结构的数值算例结果显示本文提出的算法和常规有限元相比误差小于1%,说明本文方法具有很高的精度,且由于其单元积分时无需对积分函数或者积分域进行三角化等额外处理,该方法具有很高的效率。

A new method is developed to deal with domain integrals in boundary element method(BEM) with the polygonal mesh approach for two dimensional variable coefficient problems. Firstly, the shape functions of irregular polygonal mesh are constructed for the interpolation of geometry and physical quantities. Then, the domain integrals over polygonal meshes are converted into line integrals along the polygon perimeter by employing the radial integral method. This method results in a unified scheme to evaluate various polygonal mesh integrals. Finally, three representative numerical examples on the functionally graded materials and structures demonstrate that the relative error of the results calculated by the proposed method is less than 1% by compared with those calculated by the standard finite element method. It can be concluded that the proposed method may achieve satisfied accuracy and efficiency since it does not need any special technique to the integrands or integral domains.