论文标题
适用于3D Stokes流量的弱奇异内核的新型杂交正交方案
New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow
论文作者
论文摘要
这项工作提出了四种新型杂交正交方案，以在任意平滑表面上对弱奇异边界积分（1/r内核）进行有效，准确的评估。这些积分出现在几个部分微分方程的边界元素分析中，包括粘性流的Stokes方程和声学的Helmholtz方程。提出的正交方案将基于DUFFY转换的正交规则应用于包含奇异性和经典高斯正交的表面元素，以对其余的元素。四个方案中的两个还考虑了接近奇异性的元素的特殊处理，在该元素中，使用了精致的高斯正交正交和新的时刻正交规则。 考虑两种不同的球体离散化的扁平B型贴片和NURBS球体上，将杂交正交方案系统地研究了：一个精确的单点球体，具有杆子的简并控制点，近似离散化，由六个带有常规元素的斑块组成。在Stokes流的边界元素分析中进一步证明了正交方案的效率，其中在旋转和翻译弯曲物体的稳定问题中都在收敛研究中的，网格和正交细化。与经典方案相比，提出的新方案观察到了更高的收敛率。
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.
