中文摘要：

常平均曲率曲面经常作为界面出现在许多物理问题当中,是物理膜泡的一种数学抽象,而细分曲面的灵活性及其高质量的特性使得它成为一种强有力的曲面设计工具.通过给定边界,使用由一个二阶能量范函导出的四阶几何偏微分方程和一个二阶几何偏微分方程来构造常平均曲率细分曲面,这2个方程采用有限元方法求解;由于扩展的Loop细分规则能处理带边界的曲面问题,所以采用其极限形式作为有限元空间.实验结果显示,采用文中方法能够近似地构造出具有任意拓扑结构控制网格和任意形状边界的常平均曲率曲面.

英文摘要：

Surfaces with constant mean curvature always arise as interfaces in many physical problems,and are the mathematical abstraction of physical soap films and soap bubbles.The flexibility and high quality of subdivision surfaces make them to be a powerful tool for designing surfaces.In this paper,we construct the constant mean curvature subdivision surfaces with given boundaries using a forth-order geometric partial differential equation deduced from a second-order energy functional and a second-order geometric partial differential equation.These equations are solved by a finite element method.We adopt the limit functions of the extended Loop＇s subdivision scheme as the finite element space because this scheme can treat surfaces with boundaries.The constant mean curvature subdivision surfaces can be approximately constructed with any topology of the control mesh and any shaped boundaries.