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Axioms 2018, 7(4), 76; https://doi.org/10.3390/axioms7040076

On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem

1
Department of Mathematics, Universidad Tecnológica de Pereira, 660003 Pereira, Colombia
2
Fakultät für Luft- und Raumfahrttechnik, Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany
3
Lavrentyev Institute of Hydrodynamics, Siberian Division of the Russian Academy of Sciences, Novosibirsk 630090, Russia
*
Author to whom correspondence should be addressed.
Current address: Institute for Mathematics and Scientific Computing, Karl-Franzens University of Graz, NAWI Graz, Heinrichstr, 36, 8010 Graz, Austria.
Received: 26 September 2018 / Revised: 23 October 2018 / Accepted: 25 October 2018 / Published: 27 October 2018
(This article belongs to the Special Issue Applications of Differential Equations and Dynamical Systems)
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Abstract

This paper establishes the shape derivative of geometry-dependent objective functions for use in constrained variational problems. Using a Lagrangian approach, our differentiablity result is based on the theorem of Delfour–Zolésio on directional derivatives with respect to a parameter of shape perturbation. As the key issue of the paper, we analyze the bijection under the kinematic transport of geometries that is needed for function spaces and feasible sets involved in variational problems. Our abstract theoretical result is applied to the Brinkman flow problem under incompressibility and mixed Dirichlet–Neumann boundary conditions, and provides an analytic formula of the shape derivative based on the velocity method. View Full-Text
Keywords: constrained optimization; variational inequality; Lagrangian; geometry-dependent objective function; shape derivative; Delfour–Zolésio theorem; divergence-free Brinkman flow constrained optimization; variational inequality; Lagrangian; geometry-dependent objective function; shape derivative; Delfour–Zolésio theorem; divergence-free Brinkman flow
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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González Granada, J.R.; Gwinner, J.; Kovtunenko, V.A. On the Shape Differentiability of Objectives: A Lagrangian Approach and the Brinkman Problem. Axioms 2018, 7, 76.

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