Abstract A new approach, radial integration polygonal boundary element method (RIPBEM), for solving heat conduction problems is presented in this paper. The proposed RIPBEM is a new concept in boundary element method (BEM), which would be of great flexibility in mesh generation of complex 3D geometries. Due to the characteristic of arbitrary shapes of polygonal elements, conventional shape functions are insufficient. Moreover, the resulted surface boundary integrals cannot be directly evaluated by the standard Gauss quadrature. To solve these problems, general shape functions for polygonal elements with arbitrary number of nodes are given. To generally and numerically calculate the resulted surface integrals, the radial integration method (RIM) is employed to convert the surface boundary integrals into equivalent contour line integrals of the polygonal elements. As for 3D domain integrals, they are transformed to equivalent line integrals using RIM twice. This methodology can explicitly eliminate strong singularities. Several numerical examples are given to show the effectiveness and the accuracy of the proposed polygonal boundary element method for solving heat conduction problems.

中文翻译：

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求解热传导问题的一种新的径向积分多边形边界元方法

摘要 本文提出了一种求解热传导问题的新方法——径向积分多边形边界元法(RIPBEM)。所提出的 RIPBEM 是边界元法 (BEM) 中的一个新概念，它将在复杂 3D 几何的网格生成中具有很大的灵活性。由于多边形元素具有任意形状的特性，常规的形状函数是不够的。此外，所得到的表面边界积分不能通过标准高斯求积法直接计算。为了解决这些问题，给出了具有任意节点数的多边形单元的一般形状函数。为了一般地和数值地计算结果表面积分，采用径向积分法（RIM）将表面边界积分转化为多边形单元的等效等高线积分。对于 3D 域积分，它们使用 RIM 两次转换为等效线积分。这种方法可以明确地消除强奇点。给出了几个数值例子来说明所提出的多边形边界元方法解决热传导问题的有效性和准确性。