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First variation of the generalcurvature-dependent surface energy

Published online by Cambridge University Press:  22 July 2011

Günay Doğan and
Ricardo H. Nochetto
Günay Doğan
Affiliation:
Theiss Research, San Diego, CA, USA, and Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, 20899 MD, USA. gunay.dogan@nist.gov.
Ricardo H. Nochetto
Affiliation:
Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, 20742 MD, USA. rhn@math.umd.edu.

Abstract

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We consider general surface energies, which areweighted integrals over a closed surface with a weight functiondepending on the position, the unit normal andthe mean curvature of the surface. Energiesof this form have applications in many areas, such as materials science,biology and image processing. Often one is interested in findinga surface that minimizes such an energy, which entails finding its firstvariation with respect to perturbations of the surface.We present a concise derivation of the first variation of thegeneral surface energy using tools from shape differential calculus.We first derive a scalar strong form and nexta vector weak form of the first variation. The latter reveals thevariational structure of the first variation, avoids dealingexplicitly with the tangential gradient of the unit normal,and thus can be easily discretized using parametric finite elements.Our results are valid for surfaces in any number of dimensionsand unify all previous results derived for specific examples ofsuch surface energies.

Keywords

Surface energygradient flowmean curvatureWillmore functional

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 1 , January 2012, pp. 59 - 79
Copyright
© EDP Sciences, SMAI, 2011

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