IgA-BEM for 3D Helmholtz problems on multi-patch domains using B-spline tailored numerical integration

TL;DR

This paper introduces an IgA-BEM approach for 3D Helmholtz problems on multi-patch domains, employing tailored numerical integration techniques that ensure accurate and efficient solutions with optimal convergence.

Contribution

It develops a novel IgA-BEM framework with specialized quadrature rules and an automatic threshold strategy for singularity handling, extending spline quasi-interpolation to multi-patch geometries.

Findings

Achieves expected convergence orders with uniform discretization.

Uses a small number of quadrature nodes for accurate results.

Extends singularity extraction to multi-patch settings.

Abstract

An Isogeometric Boundary Element Method (IgA-BEM) is considered for the numerical solution of Helmholtz problems on 3D bounded or unbounded domains, admitting a smooth conformal multi-patch representation of their finite boundary surface. The discretization space is formed by $C^0$ inter-patch continuous basis functions whose restriction to a patch simplifies to the span of tensor product B-splines composed with the given patch parameterization. For both regular and singular integration, the proposed model utilizes a numerical procedure defined on the support of each trial B-spline function, which makes possible a function--by--function implementation of the matrix assembly phase. Spline quasi-interpolation is the common ingredient of all the considered quadrature rules; in the singular case it is combined with a B-spline recursion over the spline degree and with a singularity extraction technique, extended to the multi-patch setting for the first time. A threshold selection strategy is proposed to automatically distinguish between nearly singular and regular integrals. Numerical examples on relevant benchmarks show that the expected convergence orders are achieved with uniform discretization and a small number of uniformly spaced quadrature nodes.