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 C^0 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.
Highlights
Hughes et al [33] proposed the concept of isogeometric analysis (IGA) in 2005
The early works on IGA [10,18,47] focussed on geometries modelled using Non-Uniform Rational B-Splines (NURBS) as these are widely used in computer aided design (CAD)
Using the finite element method to solve the partial differential equations (PDEs) on surfaces dates back to the seminal work by Dziuk [25], which developed a variational formulation to approximate the solution of the Laplace– Beltrami problems on two dimensional surfaces
Summary
Hughes et al [33] proposed the concept of isogeometric analysis (IGA) in 2005. The early works on IGA [10,18,47] focussed on geometries modelled using Non-Uniform Rational B-Splines (NURBS) as these are widely used in computer aided design (CAD). Stam [50] developed a method to evaluate Catmull–Clark subdivision surfaces directly without explicitly subdividing, allowing one to evaluate elements containing extraordinary vertices. Using the finite element method to solve the partial differential equations (PDEs) on surfaces dates back to the seminal work by Dziuk [25], which developed a variational formulation to approximate the solution of the Laplace– Beltrami problems on two dimensional surfaces. Previous studies [16,17] on Catmull–Clark subdivision surfaces for analysis introduce ghost degrees of freedoms for constructing basis functions in elements at boundaries. The proposed method can perform isogeometric analysis on complex geometries using Catmull–Clark subdivision discretisations. Catmull–Clark subdivision surfaces are limiting surfaces generated by successively subdividing given control meshes They are identical to uniform bi-cubic B-splines.
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