In this paper, we develop the isogeometric analysis of the dual boundary element method (IGA-DBEM) to solve the potential problem with a degenerate boundary. The non-uniform rational B-Spline (NURBS) based functions are employed to interpolate the geometry and physical function. To deal with the rank-deficiency problem due to the degenerate boundary, the hypersingular integral equation is introduced to promote the full rank for the influence matrix in the dual BEM. Finally, three numerical examples are given to verify the accuracy of our proposed method. Both circular and square domains subjected to the Dirichlet boundary condition are considered. The engineering problem containing a degenerate boundary is considered, e.g., a seepage flow problem with a sheet pile. Numerical results of the IGA-DBEM agree well with these of the exact solution and the original dual boundary element method.

中文翻译：

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具有退化边界的拉普拉斯问题的双边界元方法的等几何分析

在本文中，我们开发了双边界元方法 (IGA-DBEM) 的等几何分析，以解决退化边界的潜在问题。采用基于非均匀有理 B 样条 (NURBS) 的函数来插值几何和物理函数。为了解决退化边界导致的秩亏问题，引入了超奇异积分方程来提升对偶边界元影响矩阵的满秩。最后，给出了三个数值例子来验证我们提出的方法的准确性。受 Dirichlet 边界条件影响的圆形和方形域都被考虑。考虑包含退化边界的工程问题，例如板桩的渗流问题。