The second-order shape derivative of Kohn-Vogelius-type cost functional using the boundary differentiation approach.(English)Zbl 1321.49073
The article studies a class of two-dimensional boundary value problems of Bernoulli type through shape optimization methods. The cost functional is of Kohn-Vogelius type, and it is minimized over a class of admissible domains subject to two boundary value problems. An analysis of the shape derivatives of the cost functional is carried out. First, the authors introduce necessary tools such as a perturbed domain, a perturbation of identity operator, and the method of mapping. The domain and the boundary transformation formulas, as well as formulas of tangential calculus are provided next. The structure of the second-order Eulerian derivative of a general shape functional is discussed. The discussion is followed by a formal derivation of the second-order shape derivative of the functional which is the authors’ main contribution. The approach used in the computation is based on a method which makes use of the boundary differentiation formula presented in [M. C. Delfour andJ.-P. Zolésio, Shapes and geometries. Analysis, differential calculus, and optimization. Advances in Design and Control 4. Philadelphia, PA: SIAM (2001;Zbl 1002.49029)]. The authors show that the computed shape derivative has a symmetric and a nonsymmetric part, and satisfies a structure theorem. Finally, the second-order Eulerian derivative of the functional at the solution of the Bernoulli problem is discussed and the obtained formulas are compared to an existing result.
Reviewer: Baasansuren Jadamba (Rochester)
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 35R35 | Free boundary problems for PDEs |
| 35N25 | Overdetermined boundary value problems for PDEs and systems of PDEs |
Keywords:
Bernoulli problem;boundary value problems;shape optimization;shape derivative;boundary differentiationCitations:
Zbl 1002.49029Cite
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