Desingularization in boundary element analysis of three‐dimensional Stokes flow

Harmel, Maximilian; Rajski, Michal P.; Sauer, Roger A.

doi:10.1002/pamm.201800479

Cited by 4 publications

(1 citation statement)

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“…This approach is referred to as nonsingular, regularized or desingularized BEM. It is applied to linear elasticity (Scott et al, 2013;Taus et al, 2019), to Stokes flow (Taus et al, 2016;Harmel et al, 2018) and to the Helmholtz equation (Simpson et al, 2014;Peake et al, 2015), among others. Klaseboer et al (2012) further apply nonsingular BEM to fluid mechanics by considering Stokes equations for viscous flow, Laplace equation for potential flow and Helmholtz equation for free-streamline flow.…”

Section: Introductionmentioning

confidence: 99%

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

Harmel¹,

Sauer²

2022

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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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