发表机构
CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris; Mathematical Institute, University of Oxford; Mathematical Institute, Faculty of Mathematics and Physics, Charles University; School of Computing and Mathematical Sciences, University of Leicester; Physics Department, Universitat Politècnica de Catalunya; International Centre for Numerical Methods in Engineering (CIMNE)（巴黎综合理工学院; 牛津大学数学研究所; 布拉格查理大学数学与物理学院数学研究所; 莱斯特大学计算与数学科学学院; 加泰罗尼亚理工大学物理系; 国际工程数值方法中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种基于形状优化的方法，通过扩展理论处理非唯一解，并利用隐函数定理，实现对超弹性超材料突跳能量跳变的精确控制。
AI 中文摘要
多稳态系统在相同的载荷和边界条件下允许存在多个不同的稳定平衡态。多稳态结构的形状优化为设计先进材料和超材料的力学响应提供了一条有前景的途径。在本工作中，我们开发了一种公式和算法，通过优化域的形状来控制超弹性超材料中与突跳行为相关的能量跳变。关键挑战在于经典形状优化理论假设域到解映射是单值的，而这在多稳态情形下并不成立。我们通过将该理论扩展到具有非唯一解的情形来解决这一问题，将域变形的缩放视为连续参数，并应用隐函数定理给出随着形状变化多个解分支持续存在的充分条件。这一理论基础支撑了一种实用的算法，该算法以在稳定态之间实现预定的能量跳变为目标，从而能够系统性地设计其突跳响应可按需调谐的结构。
英文摘要
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.
