New hybrid quadrature schemes for weakly singular kernels applied to isogeometric boundary elements for 3D Stokes flow

TL;DR

This paper introduces four innovative hybrid quadrature schemes designed for efficient and precise evaluation of weakly singular boundary integrals in boundary element methods, significantly improving convergence rates for 3D Stokes flow problems.

Contribution

The paper presents four new hybrid quadrature schemes combining Duffy transform and Gaussian quadrature, with special treatments near singularities, tailored for boundary element analysis of 3D Stokes flow.

Findings

Higher convergence rates compared to classical schemes.

Effective handling of singularities on smooth surfaces.

Validated on flat B-spline patches and NURBS spheres.

Abstract

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.