Isogeometric Methods for Free Boundary Problems

TL;DR

This paper develops and compares three isogeometric algorithms for free boundary problems, demonstrating that collocation methods are robust and efficient, with potential applications to fluid dynamics.

Contribution

It introduces three quasi-Newton isogeometric algorithms for free boundary problems, highlighting the advantages of collocation schemes over Galerkin methods.

Findings

Collocation method is more robust and efficient.

Isogeometric analysis accurately models curved geometries.

Algorithms perform well on benchmark tests.

Abstract

We present in detail three different quasi-Newton isogeometric algorithms for the treatment of free boundary problems. Two algorithms are based on standard Galerkin formulations, while the third is a fully-collocated scheme. With respect to standard approaches, isogeometric analysis enables the accurate description of curved geometries, and is thus particularly suitable for free boundary numerical simulation. We apply the algorithms and compare their performances to several benchmark tests, considering both Dirichlet and periodic boundary conditions. In this context, iogeometric collocation turns out to be robust and computationally more efficient than Galerkin. Our results constitute a starting point of an in-depth analysis of the Euler equations for incompressible fluids.