An adaptive IgA‐BEM with hierarchical B‐splines based on quasi‐interpolation quadrature schemes

Abstract

Summary The isogeometric formulation of the boundary element method (IgA‐BEM) is investigated within the adaptivity framework. Suitable weighted quadrature rules to evaluate integrals appearing in the Galerkin BEM formulation of 2D Laplace model problems are introduced. The proposed quadrature schemes are based on a spline quasi‐interpolation (QI) operator and properly framed in the hierarchical setting. The local nature of the QI perfectly fits with hierarchical spline constructions and leads to an efficient and accurate numerical scheme. An automatic adaptive refinement strategy is driven by a residual‐based error estimator. Numerical examples show that the optimal convergence rate of the Galerkin solution is recovered by the proposed adaptive method.

A Study on Spline Quasi-interpolation Based Quadrature Rules for the Isogeometric Galerkin BEM

“…To obtain a regular K 1 , the function δ needs to be chosen according to the type of domain. Following the idea from [14], it is defined as:…”

“…In this section we summarise the two spline quasi-interpolation based quadrature rules introduced in [1]. The quadratures are adopted to evaluate regular and singular integrals that appear in the system matrix (12) and in the right-hand side vector (13)- (14). Some implementation aspects are explained afterwards.…”

“…To evaluate entries in (12)- (14), the computation of the following two types of integrals needs to be addressed,…”

“…Explicit formulae to compute the moments are derived in [11,1] for δ = |s − t|. We refer to [14], when δ takes the second form in (11), suitable for a closed boundary curve. The procedure 2 differs from the procedure 1 in the initial step, where only the function g is approximated by a QI spline σ g in the local spaceŜ τ of degree p:…”

“…In particular, the derived quadrature formulae are obtained using a quasi-interpolation (QI ) operator, firstly introduced in [12] and then applied to construct quadrature rules for regular integrals in [13]. The second procedure has been successfully applied in a Galerkin adaptive BEM using hierarchical B-splines in [14]. The authors also provide some theoretical results about the convergence order of the quadrature rule, when h-refinement is performed.…”