An Isogeometric Boundary Element Method with Subdivision Surfaces for Helmholtz analysis
Zhaowei Liu  1, *@  , Robert Simpson  1@  , Fehmi Cirak  2@  
1 : School of Engineering
University of Glasgow, Glasgow G12 8QQ -  United Kingdom
2 : Department of Engineering
University of Cambridge, Trumpington Street, Cambridge CB2 1PZ -  United Kingdom
* : Corresponding author

Isogeometric Analysis (IGA) has rapidly expanded in recent years into a major research effort to link Computer Aided Design (CAD) and numerical methods driven by the need for more effcient industrial design tools. The central idea of IGA is that the same discretisation model is used for both design and analysis which eliminates costly model conversion processes encountered in traditional engineering design work flows. IGA was originally conceived by Hughes et al. [1] and has predominately focussed on the use of the Finite Element Method, but work has also applied the approach to the Boundary Element Method (BEM) where distinct advantages are found, stemming from the need for only a surface mesh. In the majority of IGA implementations the most commonly used CAD discretisation is the Non-Uniform Rational B-Spline (NURBS) due to its ubiquitous nature in CAD software. However, NURBS technology has limitations due to its tensor-product nature and a number of researchers have developed alternative CAD discretisations which overcome this limitation. One such example is T-spline technology which has been employed in an IGA setting by Bazilevs et al. [2] in 2010. Subdivision surfaces provide another alternative for overcoming the limitations of NURBS, initially introduced by Cirak [3] in 2000.

The present work develops a new isogeometric Boundary Element Method with subdivision surfaces for solving Helmholtz problems. We find that by adopting the high order (quartic) basis functions of subdivision surfaces to perform Helmholtz analysis a higher accuracy per degree of freedom is obtained over equivalent Lagrangian discretisations. We demonstrate this through two Helmholtz problems with closed-form solutions: a pulsating sphere and a plane wave impinged on a `hard' sphere. Furthermore, we illustrate how the approach is able to handle geometries with arbitrary complexity showcasing the potential of the approach for integrated design and analysis software.

[1] Hughes et al., CMAME 194 (39) pp.4135-4195 (2005)

[2] Bazilevs et al. CMAME 199 (5-8) pp. 229-263 (2010) 

[3] Cirak et al., IJNME 47 (12) pp.2039-2072 (2000) 

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