Abstract
We derive the second-order shape derivatives (shape Hessians) of cost functions for shape optimization problems of domains in which boundary value problems of partial differential equations are defined, and propose an \(H^1\) Newton method to solve the problems using the shape Hessians. In this paper, we formulate an abstract shape optimization problem and show the computations of the first- and second-order shape derivatives of cost functions under the abstract framework. Then, using the shape gradients and Hessians, we propose an \(H^1\) Newton method to solve the given problem. As an illustration, the shape Hessians of a mean compliance and a domain measure are derived and then used for a numerical example.
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References
Azegami, H.: Shape Optimization Problems (in Japanese). Morikita Publishing, Tokyo (2016)
Azegami, H.: Solution of shape optimization problem and its application to product design. Mathematics for Industry 2017, vol. 26, pp. 83–98. Springer, Singapore (2016)
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Acknowledgements
The present study was supported by JSPS KAKENHI (16K05285).
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Azegami, H. (2018). Second Derivatives of Cost Functions and \(H^1\) Newton Method in Shape Optimization Problems. In: van Meurs, P., Kimura, M., Notsu, H. (eds) Mathematical Analysis of Continuum Mechanics and Industrial Applications II. CoMFoS 2016. Mathematics for Industry, vol 30. Springer, Singapore. https://doi.org/10.1007/978-981-10-6283-4_6
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DOI: https://doi.org/10.1007/978-981-10-6283-4_6
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