[1] U. Clarenz, U. Diewald, G. Dziuk, M. Rumpf, and R. Rusu. A finite element method for surface restoration with smooth boundary conditions. Computer Aided Geometric Design, 21(5):427-445, 2004.
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In surface restoration usually a damaged region of a surface has to be replaced by a surface patch which restores the region in a suitable way. In particular one aims for C1-continuity at the patch boundary. The Willmore energy is considered to measure fairness and to allow appropriate boundary conditions to ensure continuity of the normal field. The corresponding L2-gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as system of two coupled second order equations. As it is well-known, fourth order problems require an implicit time discretization. Here a semi-implicit approach is presented which allows large timesteps. For the discretization of the boundary condition, two different numerical methods are introduced. Finally, we show applications to different surface restoration problems.