The Isogeometric Nystr\"om Method

TL;DR

This paper introduces the isogeometric Nyström method, a boundary integral approach that uses pointwise evaluations and Bezier elements, enabling efficient analysis of complex geometries with higher order convergence.

Contribution

It presents a novel isogeometric Nyström method that integrates boundary integral equations with isogeometric analysis, using local correction and Bezier elements for singular integrals.

Findings

Higher order convergence in 2D and 3D numerical tests

Applicable to Laplace and Lame-Navier equations

Provides a flexible alternative to existing boundary integral methods

Abstract

In this paper the isogeometric Nystr\"om method is presented. It's outstanding features are: it allows the analysis of domains described by many different geometrical mapping methods in computer aided geometric design and it requires only pointwise function evaluations just like isogeometric collocation methods. The analysis of the computational domain is carried out by means of boundary integral equations, therefor only the boundary representation is required. The method is thoroughly integrated into the isogeometric framework. For example, the regularization of the arising singular integrals performed with local correction as well as the interpolation of the pointwise existing results are carried out by means of Bezier elements. The presented isogeometric Nystr\"om method is applied to practical problems solved by the Laplace and the Lame-Navier equation. Numerical tests show higher order convergence in two and three dimensions. It is concluded that the presented approach provides a simple and flexible alternative to currently used methods for solving boundary integral equations, but has some limitations.