Incorporating topological derivatives into shape derivatives based level set methods

Authors:
Lin He;Chiu-Yen Kao;Stanley Osher
Affiliations:
UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA;Institute for Mathematics and its Applications, 400 Lind Hall, 207 Church Street, Minneapolis, MN 55455, USA;UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, USA
Venue:
Journal of Computational Physics
Year:
2007

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Abstract

Shape derivatives and topological derivatives have been incorporated into level set methods to investigate shape optimization problems. The shape derivative measures the sensitivity of boundary perturbations while the topological derivative measures the sensitivity of creating a small hole in the interior domain. The combination of these two derivatives yields an efficient algorithm which has more flexibility in shape changing and may escape from a local optimal. Examples on finding the optimal shapes for maximal band gaps in photonic crystal and acoustic drum problems are demonstrated.