This work proposes four novel hybrid quadrature schemes for the efficient and accurate evaluation of weakly singular boundary integrals (1/r kernel) on arbitrary smooth surfaces. Such integrals appear in boundary element analysis for several partial differential equations including the Stokes equation for viscous flow and the Helmholtz equation for acoustics. The proposed quadrature schemes apply a Duffy transform-based quadrature rule to surface elements containing the singularity and classical Gaussian quadrature to the remaining elements. Two of the four schemes additionally consider a special treatment for elements near to the singularity, where refined Gaussian quadrature and a new moment-fitting quadrature rule are used. The hybrid quadrature schemes are systematically studied on flat B-spline patches and on NURBS spheres considering two different sphere discretizations: An exact single-patch sphere with degenerate control points at the poles and an approximate discretization that consist of six patches with regular elements. The efficiency of the quadrature schemes is further demonstrated in boundary element analysis for Stokes flow, where steady problems with rotating and translating curved objects are investigated in convergence studies for both, mesh and quadrature refinement. Much higher convergence rates are observed for the proposed new schemes in comparison to classical schemes.

翻译：本文提出了四种新型混合求积格式，用于在任意光滑曲面上高效精确地评估弱奇异边界积分（1/r核）。这类积分出现在多个偏微分方程的边界元分析中，包括粘性流的斯托克斯方程和声学的亥姆霍兹方程。所提出的求积格式对包含奇异性的面单元应用基于Duffy变换的求积规则，而对其余单元采用经典高斯求积。其中两种格式还额外对靠近奇异性的单元进行了特殊处理，分别使用细化高斯求积和一种新的矩拟合求积规则。这些混合求积格式在平面B样条片和NURBS球面上进行了系统研究，考虑了两种不同的球面离散化：一种是在极点处具有退化控制点的精确单块球面，另一种是由六个规则单元块组成的近似离散化。通过斯托克斯流的边界元分析进一步展示了求积格式的效率，其中在网格和求积细化的收敛性研究中，对旋转和平移弯曲物体的稳态问题进行了研究。与经典格式相比，所提出的新格式观察到了更高的收敛速率。