## A finite element method for surface restoration with smooth boundary conditions.(English)Zbl 1069.65546

Summary: 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 $$C^{1}$$-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 $$L^{2}$$-gradient flow as the actual restoration process leads to a system of fourth order partial differential equations, which can also be written as a 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 time steps. For the discretization of the boundary condition, two different numerical methods are introduced. Finally, we show applications to different surface restoration problems.

### MSC:

 65D18 Numerical aspects of computer graphics, image analysis, and computational geometry 65D17 Computer-aided design (modeling of curves and surfaces) 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs

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### References:

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