Controlling the energy jump of multistable structures using shape optimization
Abstract
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.
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BibTeX
@misc{slimane2026controlling,
title = {{Controlling the energy jump of multistable structures using shape optimization}},
author = {Arselane Hadj Slimane and Patrick E. Farrell and Alberto Paganini and Àlex Ferrer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12816},
eprint = {2609.12816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12816},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Controlling the energy jump of multistable structures using shape optimization
AU - Arselane Hadj Slimane
AU - Patrick E. Farrell
AU - Alberto Paganini
AU - Àlex Ferrer
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12816
UR - https://arxiv.org/abs/2609.12816
AB - 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.
ER -