A finite element method for surface restoration with smooth boundary conditions

Authors:
U. Clarenz;U. Diewald;G. Dziuk;M. Rumpf;R. Rusu
Affiliations:
Numerische Mathematik und Wissenschaftliches Rechnen, Institut für Mathematik, Fakultät 4, Naturwissenschaften, Gerhard-Mercator-Universität Duisburg, Lotharstr. 65, D-47048 Duisbur ...;Numerische Mathematik und Wissenschaftliches Rechnen, Institut für Mathematik, Fakultät 4, Naturwissenschaften, Gerhard-Mercator-Universität Duisburg, Lotharstr. 65, D-47048 Duisbur ...;Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Str. 10, D-79104 Freiburg i. Br., Germany;Numerische Mathematik und Wissenschaftliches Rechnen, Institut für Mathematik, Fakultät 4, Naturwissenschaften, Gerhard-Mercator-Universität Duisburg, Lotharstr. 65, D-47048 Duisbur ...;Abteilung für Angewandte Mathematik, Universität Freiburg, Hermann-Herder-Str. 10, D-79104 Freiburg i. Br., Germany
Venue:
Computer Aided Geometric Design
Year:
2004

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Abstract

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 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.