Assessment of an isogeometric approach with Catmull-Clark subdivision surfaces using the Laplace-Beltrami problems

Liu, Z. , McBride, A. , Saxena, P. and Steinmann, P. (2020) Assessment of an isogeometric approach with Catmull-Clark subdivision surfaces using the Laplace-Beltrami problems. Computational Mechanics, 66, pp. 851-876. (doi: 10.1007/s00466-020-01877-3)

Abstract

An isogeometric approach for solving the Laplace–Beltrami equation on a two-dimensional manifold embedded in three-dimensional space using a Galerkin method based on Catmull–Clark subdivision surfaces is presented and assessed. The scalar-valued Laplace–Beltrami equation requires only C0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull–Clark subdivision surfaces. Catmull–Clark subdivision bases are used to discretise both the geometry and the physical field. A fitting method generates control meshes to approximate any given geometry with Catmull–Clark subdivision surfaces. The performance of the Catmull–Clark subdivision method is compared to the conventional finite element method. Subdivision surfaces without extraordinary vertices show the optimal convergence rate. However, extraordinary vertices introduce error, which decreases the convergence rate. A comparative study shows the effect of the number and valences of the extraordinary vertices on accuracy and convergence. An adaptive quadrature scheme is shown to reduce the error.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: McBride, Professor Andrew and Liu, Dr Zhaowei and Saxena, Dr Prashant and Steinmann, Professor Paul
• Authors: Liu, Z., McBride, A., Saxena, P., and Steinmann, P.
• College/School: College of Science and Engineering > School of Engineering > Infrastructure and Environment
• Journal Name: Computational Mechanics
• Publisher: Springer
• ISSN: 0178-7675
• ISSN (Online): 1432-0924
• Published Online: 15 July 2020
• Copyright Holders: Copyright © The Author(s) 2020
• First Published: First published in Computational Mechanics 66:851–876
• Publisher Policy: Reproduced under a Creative Commons license