In this paper, we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities, we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments. © 2011 World Scientific Publishing Company.
|Original language||English (US)|
|Number of pages||31|
|Journal||Mathematical Models and Methods in Applied Sciences|
|State||Published - Oct 24 2011|
Bibliographical noteKAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: Part of this work was carried out when the authors were with the Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz, and the Johannes Kepler University Linz, respectively. The authors thank Heinz Engl (RICAM and University of Vienna) and Peter Markowich (Cambridge University and RICAM) for stimulating this joint research. M.B. acknowledges financial support by the Austrian Science Foundation FWF through project SFB F 013/08, and the German Research Foundation DFG through the project Regularization with Singular Energies. Work of N.M. was partially supported by the WWTF (Wiener Wissenschafts, Forschungs und Technologiefonds) project No. CI06 003, while he was at the University of Vienna. Also the work of N.M. and M.T.W. supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
This publication acknowledges KAUST support, but has no KAUST affiliated authors.