Abstract
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape derivative for general domains is computedvia the boundary differentiation scheme. Additionally, the second-order shape derivative ofJ at the solution of the Bernoulli problem is computed using Tiihonen’s approach.
Highlights
Nowadays many researchers are interested in a class of boundary value problems called free boundary problems
Moving boundary problems are usually associated to problems that vary with time
This paper studies a class of two-dimensional free boundary problems of Bernoulli type
Summary
Nowadays many researchers are interested in a class of boundary value problems called free boundary problems These are mostly partial differential equations to be solved for both unknown state function(s) and an unknown domain. Free boundary problems are not confined only to the study of phase transitions, such as that of solidification or melting of a particular material, or to the study of fluid dynamics They arise in the study of image development in electrophotography, chemical vapor deposition, and tumor growth [3]. We are interested in the exterior Bernoulli problem (BP), which can be described as follows: Given a bounded and connected domain A ⊂ R2 with a fixed boundary Γ := ∂A and a constant λ < 0, one needs to find a bounded connected domain B ⊂ R2 with a free boundary Σ and containing the closure of A, and an associated state function u : Ω → R, where Ω = B\Ā, such that the following conditions are met:. As well as shape derivatives of the state variables, are highly involved in the approach that we used
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