Development and implementation of numerical algorithm for variational methods and generalized gradient flows for geometric evolution problems of higher order for surface processing in computer graphics

美国帕萨迪纳市加州理工学院Mathieu Desbrun教授研究小组的主要研究目标是在各向异性表面光顺和表面恢复的背景下，开发和实现各向异性Willmore演化问题的数值算法。该应用程序包括混合问题，不同的表面补丁连接的其他表面补丁定义为各向异性Willmore功能的极小化。除了混合问题，表面处理问题也被考虑。在那里，曲面的破坏区域被替换为曲面片，该曲面片以适当的方式恢复曲面。特别地，要求在面片边界处满足C1条件。由于各向异性Willmore泛函的L2-梯度流对应于一个高度非线性的抛物型偏微分方程，该泛函非常适合于这类恢复问题。选择一个不同的度量，例如H^1-度量而不是L2-度量，考虑广义梯度流。

The main goal of the research project in the research group of Prof. Mathieu Desbrun at the California Institute of Technology (Caltech) in Pasadena, USA, will be the development and implementation of numerical algorithms for evolution problems of the anisotropic Willmore in the context of anisotropic surface fairing and surface restoration. The application includes blending problems, where different surface patches are connected by other surface patches defined as minimizers of the anisotropic Willmore functional. Besides blending problems surface restorations problems are considered. There, a destroyed region of a surface is replaced by a surface patch that restores the surface in a suitable way. In particular one ask for C1-condition at the patch boundary. Since the L2-gradient flow of the anisotropic Willmore functional corresponds to a highly non linear parabolic partial differential equation, this functional is well suited for this kind of restoration problems. Choosing a different metric, e.g. the H^1-metric instead of the L2-metric, generalized gradient flows are considered.