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|