Volume 29, 2023
|Number of page(s)||35|
|Published online||11 January 2023|
An abstract Lagrangian framework for computing shape derivatives
Instituto de Matemática e Estatística, Universidade de São Paulo,
Rua do Matão 1010,
2 Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Avenida Trabalhador São-Carlense, 400, São Carlos, SP, Brazil
* Corresponding author: firstname.lastname@example.org
Accepted: 24 November 2022
In this paper we study an abstract framework for computing shape derivatives of functionals subject to PDE constraints in Banach spaces. We revisit the Lagrangian approach using the implicit function theorem in an abstract setting tailored for applications to shape optimization. This abstract framework yields practical formulae to compute the derivative of a shape functional, the material derivative of the state, and the adjoint state. Furthermore, it allows to gain insight on the duality between the material derivative of the state and the adjoint state. We show several applications of this method to the computation of distributed shape derivatives for problems involving linear elliptic, nonlinear elliptic, parabolic PDEs and distributions. We also compare our approach with other techniques for computing shape derivatives including the material derivative method and the averaged adjoint method.
Mathematics Subject Classification: 49Q10 / 49Q12 / 35Q93 / 35R37
Key words: Shape optimization / Lagrangian methods / distributed shape derivatives
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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