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

Abstract

To cite this version:Fabien Caubet, Djalil Kateb, Frédérique Le Louër. Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell type -Application to compliance minimization. Journal of Elasticity, Springer Verlag, In press, Journal of Elasticity. <hal-01525249> 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 structure 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.

Shape Sensitivity of Eigenvalue Functionals for Scalar Problems: Computing the Semi-derivative of a Minimum

“…The models considered here, in addition to including well known and classical eigenvalue problems for the Laplacian spectrum with Dirichlet or Neumann boundary conditions, apply to completely new situations such as the Laplacian eigenvalue problem for the Wentzell boundary condition which is motivated by coating problems. We would like to mention that the Wentzell boundary condition is a natural asymptotic boundary condition on the limiting domain when a domain has a thin outer layer (see [17,18,27,28] and also [4]). With modern methods of manufacturing such as 3-d printing it is now quite easy to fabricate material pieces with a thin coating or with a spatially varying filling.…”

Shape Derivative for Some Eigenvalue Functionals in Elasticity Theory

“…Wentzell boundary conditions. Let us emphasize that such boundary conditions are not a mathematical curiosity but appear naturally in the context of linear elasticity as soon as the configuration presents discontinuities on the material properties on a submanifold (see, e.g., [9] for a crusted body or [20] for an interface problem).…”

“…In particular, the Wentzell boundary conditions, coming from asymptotic analysis (see [20,21,9] for the mechanical and theoretical justification of such conditions), permit to model coating or membrane effects. Notice that this approximation of an original structure with a thin layer by adhering to another domain with new boundary conditions, called generalized impedance boundary conditions, is a classical method in order to avoid huge difficulties in the theoretical and numerical analysis of a thin structure (for instance a mesh refinement adapted to the thickness of the layer).…”

“…If we take β = 0 and α = 0 in (2.5), we obtain the Steklov eigenvalue problem. Finally, for the choice β > 0, we have the Wentzell eigenvalue problem (see [9] for the model and the derivation of the Wentzell boundary conditions in the elasticity case).…”