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

位于美国帕萨迪纳的加州理工学院（Caltech）的Mathieu Desbrun教授的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，其中不同的表面斑块由定义为各向异性Willmore泛函的最小化的其他表面斑块连接。除了混合问题外，还考虑了表面恢复问题。在那里，一个表面的破坏区域被一个表面补丁取代，以一种合适的方式恢复表面。特别是在patch边界处要求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，因此该泛函非常适合于这类恢复问题。选择一个不同的度量，例如用H^1度量代替l2度量，可以考虑广义梯度流。

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.