A semi-analytical treatment for nearly singular integrals arising in the isogeometric boundary element method-based solutions of 3D potential problems

Han, Zhilin; Pan, Wei; Chen, Cheng; Hu, Zongjun; Niu, Zhongrong

doi:10.1016/j.cma.2022.115179

Cited by 8 publications

(1 citation statement)

References 62 publications

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“…The quadrature scheme presented in Gong et al (2020) combines the benefits of the sinh transformation method and adaptive methods such it is capable of integral kernels of the type 1/r, 1/r 2 and 1/r 3 on near singular elements. Many other quadrature rules for weakly singular integrals on nearly singular elements can be found in the literature as for example the parameterized Gaussian quadrature (Lutz, 1992), line integral approaches (Krishnasamy et al, 1994;Liu, 1998) and analytical (Padhi et al, 2001;Zhou et al, 2008) and semi-analytical methods (Sladek et al, 2001;Niu et al, 2005;Han et al, 2022).…”

Section: Approximation Of Weakly Near Singular Integralsmentioning

confidence: 99%

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

Harmel¹,

Sauer²

2022

Preprint

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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 (Duffy, 1982) 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.

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