Assessment of an Isogeometric Approach with Catmull-Clark Subdivision Surfaces using the Laplace-Beltrami Problems

TL;DR

This paper evaluates an isogeometric method using Catmull-Clark subdivision surfaces for solving the Laplace-Beltrami equation on 2D manifolds, highlighting convergence properties and the impact of extraordinary vertices.

Contribution

It introduces an isogeometric approach with Catmull-Clark subdivision surfaces for Laplace-Beltrami problems and compares its performance to finite element methods.

Findings

Subdivision surfaces without extraordinary vertices achieve optimal convergence.

Extraordinary vertices cause errors and reduce convergence rate.

Adaptive quadrature reduces approximation errors.

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.