A Smoothly Varying Quadrature Approach for 3D IgA-BEM Discretizations: Application to Stokes Flow Simulations

TL;DR

This paper presents a new quadrature method for 3D IgA-BEM that adaptively handles singularities and improves accuracy and efficiency in Stokes flow simulations.

Contribution

It introduces a smoothly varying, adaptive quadrature strategy that eliminates the need for integral classification, enhancing robustness and reducing computational costs in IgA-BEM.

Findings

Achieves excellent convergence rates in 3D Stokes problem benchmarks.

Reduces computational cost by integrating over B-spline supports.

Enhances accuracy and robustness through automatic calibration of the quadrature.

Abstract

We introduce a novel quadrature strategy for Isogeometric Analysis (IgA) boundary element discretizations, specifically tailored to collocation methods. Thanks to the dimensionality reduction and the natural handling of unbounded domains, boundary integral formulations are particularly appealing in the IgA framework. However, they require the evaluation of boundary integrals whose kernels exhibit singular or nearly singular behavior. Even when the kernel is not singular, its numerical evaluation becomes challenging whenever the integration region lies close to a collocation point. These integrals of polar and nearly singular functions represent the main computational difficulty of IgA-BEM and motivate the development of efficient and accurate quadrature rules. Unlike traditional methods that classify integrals as singular, nearly singular, or regular, our approach employs a desingularizing change of variables that smoothly adapts to the physical distance from singularities in the boundary integral kernels. The transformation intensifies near the polar point and progressively weakens when integrating over portions of the domain that are farther from it, ultimately leaving the integrand unchanged in the limit of a diametrically opposed region. This automatic calibration enhances accuracy and robustness by eliminating the traditional classification step, to which the approximation quality is often highly sensitive. Moreover, integration is performed directly over B-spline supports rather than over individual elements, reducing computational cost, particularly for higher-degree splines. The proposed method is validated through boundary element benchmarks for the three dimensional Stokes problem, where we achieve excellent convergence rates.