Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation

Julius Fergy T. Rabago; Jerico B. Bacani

doi:10.3934/cpaa.2018127

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2018, Volume 17, Issue 6: 2683-2702. Doi: 10.3934/cpaa.2018127

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Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation

Julius Fergy T. Rabago and
Jerico B. Bacani^,

Department of Mathematics and Computer Science, College of Science, University of the Philippines Baguio, Gov. Pack Rd., Baguio City 2600, Philippines

* Corresponding author
* Corresponding author

Received: February 28, 2017

Revised: July 31, 2017

Published: June 2018

This work was an output from a project funded by the UP System Emerging Interdisciplinary Research (EIDR) Program (OVPAA-EIDR-C05-015).

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Abstract

The exterior Bernoulli free boundary problem is considered and reformulated into a shape optimization setting wherein the Neumann data is being tracked. The shape differentiability of the cost functional associated with the formulation is studied, and the expression for its shape derivative is established through a Lagrangian formulation coupled with the velocity method. Also, it is illustrated how the computed shape derivative can be combined with the modified $H^1$ gradient method to obtain an efficient algorithm for the numerical solution of the shape optimization problem.

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Mathematics Subject Classification: Primary: 35R35, 35N25; Secondary: 49K20, 49Q10.

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Figure 1. Initial and final shape of the annular domain $\Omega$

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Figure 2. Initial and final shape of the annular domain $\Omega$

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Figure 3. History of values of the cost functional $J$

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Table 1. Cost Values

Iter. Cost
(α = 0.001) Cost
(α = 0.01)

1 128.187510 128.187510

2 0.19825995 0.36967179

3 0.06116257 0.00004442

4 0.00000512 0.00000004

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