# Efficient assembly based on B-spline tailored quadrature rules for the IgA-SGBEM

## TL;DR

This paper introduces B-spline tailored quadrature rules within Isogeometric Analysis for Boundary Element Methods, significantly improving computational efficiency and accuracy in assembling linear systems for 2D elliptic problems.

## Contribution

It develops new weighted quadrature schemes tailored for B-splines, enhancing the assembly process in IgA-SGBEM by reducing computational cost and increasing accuracy.

## Key findings

- New quadrature rules are exact for the test space including singular kernels.
- The proposed method reduces assembly time compared to standard element-by-element approaches.
- Simulations demonstrate improved accuracy and efficiency in integral evaluation.

## Abstract

This paper deals with the discrete counterpart of 2D elliptic model problems rewritten in terms of Boundary Integral Equations. The study is done within the framework of Isogeometric Analysis based on B-splines. In such a context, the problem of constructing appropriate, accurate and efficient quadrature rules for the Symmetric Galerkin Boundary Element Method is here investigated. The new integration schemes, together with row assembly and sum factorization, are used to build a more efficient strategy to derive the final linear system of equations. Key ingredients are weighted quadrature rules tailored for B--splines, that are constructed to be exact in the whole test space, also with respect to the singular kernel. Several simulations are presented and discussed, showing accurate evaluation of the involved integrals and outlining the superiority of the new approach in terms of computational cost and elapsed time with respect to the standard element-by-element assembly.