基于仿射变换的奇异积分自适应单元细分法

AN ADAPTIVE ELEMENT SUBDIVISION METHOD FOR SINGULAR INTEGRALS BASED ON AFFINE TRANSFORMATIONS

  • 摘要: 边界元法已广泛应用于解决工程实际问题中,精确高效地计算奇异积分对于求解边界积分方程至关重要。为此,提出了一种基于仿射变换的奇异积分自适应单元细分法,用于解决边界积分方程中的奇异性问题。其基本思想是通过仿射变换对单元进行特征分区,并结合自适应二叉树细分技术生成高质量的积分单元。利用仿射变换和特征分区技术将初始单元划分为细分区域和投影区域,不同区域的单元细分算法分别执行、计算效率高。源点附近的细分子块采用Serendipity积分单元构建,其余积分单元则采用传统积分方法进行计算。与传统单元细分方法相比,该方法具有自适应细分、积分精度高、易于实现等优点。数值算例验证了该文方法具有良好的准确性、鲁棒性和可行性。

     

    Abstract: Boundary element method (BEM) has been widely used for solving the practical engineering problems. Accurate and efficient evaluation of singular integral is of crucial importance for solving the boundary integral equations. In the BEM implementation, an adaptive element subdivision method for singular integrals based on affine transformations is presented for evaluating the singular integrals. The basic idea consists of partitioning the input surface element via affine transformation and then generating a set of high-quality patches by adaptive binary-tree subdivision. By using the domain partitioning technique, the surface element can be divided into several element projection and subdivision regions under affine transformations. It is far more efficient to separately perform the element subdivision for different regions where the desirable patches are required. The ultimate sub-elements in the vicinity of the singular point are constructed by the serendipity patches, while the remaining patches are evaluated accurately by the conventional quadrature techniques. The proposed method has some advantages over the conventional element subdivision methods, such as the adaptive element subdivision, the improved accuracy and the straight-forward implementation. Numerical results are provided to validate the accuracy, robustness and availability of the proposed method.

     

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