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.

中文翻译：

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使用 Laplace-Beltrami 问题评估 Catmull-Clark 细分曲面的等几何方法

提出并评估了使用基于 Catmull-Clark 细分曲面的 Galerkin 方法在嵌入三维空间的二维流形上求解 Laplace-Beltrami 方程的等几何方法。标量值 Laplace-Beltrami 方程只需要 $$C^0$$ 连续性，并被用来阐明使用 Catmull-Clark 细分曲面的等几何方法的关键特征和属性。Catmull-Clark 细分基用于离散几何和物理场。拟合方法生成控制网格以使用 Catmull-Clark 细分曲面近似任何给定的几何图形。Catmull-Clark 细分方法的性能与传统的有限元方法进行了比较。没有异常顶点的细分曲面显示最佳收敛速度。然而，异常顶点会引入误差，从而降低收敛速度。一项比较研究显示了异常顶点的数量和价数对准确性和收敛性的影响。显示了一种自适应正交方案以减少误差。