Controlling the energy jump of multistable structures using shape optimization

Multistable systems admit distinct stable equilibria for the same loading and boundary conditions. Shape optimization of multistable structures offers a promising route to engineer the mechanical response of advanced materials and metamaterials. In this work, we develop a formulation and algorithm for controlling the energy jump associated with snap-through behaviour in hyperelastic metamaterials by optimizing the shape of the domain. The key challenge is that classical shape optimization theory assumes a single-valued domain-to-solution map, which does not hold in the multistable setting. We address this by extending the theory to settings with non-unique solutions, treating the scaling of the domain deformation as a continuation parameter and applying the implicit function theorem to give sufficient conditions for the continued existence of multiple solution branches as the shape varies. This theoretical foundation underpins a practical algorithm that targets a prescribed energy jump between stable states, enabling the systematic design of structures whose snap-through response can be tuned on demand.