Optimal Elastostatic Cloaks

TL;DR

This paper develops a method for designing optimal elastic cloaks using the adjoint state method, enabling effective hiding of inhomogeneities in elastic materials through tailored elastic moduli.

Contribution

It introduces a novel optimization framework for elastic cloaks based on the adjoint state method with penalization for smoothness, including numerical examples for various geometries and loads.

Findings

Optimal cloaks can be designed for different geometries and load conditions.

Auxetic materials may be optimal for cloaking cavities.

The method ensures smooth inhomogeneous elastic moduli within the cloak.

Abstract

An elastic cloak hides a hole or an inhomogeneity from elastic fields. In this paper, a formulation of the optimal design of elastic cloaks based on the adjoint state method, in which the balance of linear momentum is enforced as a constraint, is presented. The design parameters are the elastic moduli of the cloak, while the objective function is a measure of the distance between the solutions in the physical and in the virtual bodies. Both the elastic medium and the cloak are assumed to be made of isotropic linear elastic materials. In order to guarantee smooth inhomogeneous elastic moduli within the cloak a penalization term is added to the objective function. Mixed finite elements are used for discretizing the weak formulation of the optimization problem. Several numerical examples of optimal elastic cloaks designed for both single and multiple loads are presented. We consider different geometries and loading types and observe that in some cases the optimal elastic cloaks for cloaking holes (cavities) are made of auxetic materials.