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Journal of Elasticity

, Volume 136, Issue 1, pp 17–53

# Shape Sensitivity Analysis for Elastic Structures with Generalized Impedance Boundary Conditions of the Wentzell Type—Application to Compliance Minimization

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## Abstract

This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structures is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations.

## Keywords

Asymptotic analysis Generalized impedance boundary conditions Wentzell conditions Shape calculus Shape sensitivity analysis Compliance minimization Linear elasticity

## Mathematics Subject Classification

35C20 49Q10 49Q12 74B05 74P05

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## Copyright information

© Springer Nature B.V. 2018

## Authors and Affiliations

• Fabien Caubet
• 1
• Djalil Kateb
• 2
• Frédérique Le Louër
• 2
Email author
1. 1.Institut de Mathématiques de ToulouseUniversité de ToulouseToulouse Cedex 9France
2. 2.LMAC EA2222 Laboratoire de Mathématiques Appliquées de Compiègne, Sorbonne UniversitésUniversité de Technologie de CompiègneCompiègne cedexFrance