  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.
