Identifying material geometries that lead to metamaterials with desired functionalities presents a challenge for the field. Discrete, or reduced-order, models provide a concise description of complex phenomena, such as negative refraction, or topological surface states; therefore, the combination of geometric building blocks to replicate discrete models presenting the desired features represents a promising approach. However, there is no reliable way to solve such an inverse problem. Here, we introduce ‘perturbative metamaterials’, a class of metamaterials consisting of weakly interacting unit cells. The weak interaction allows us to associate each element of the discrete model with individual geometric features of the metamaterial, thereby enabling a systematic design process. We demonstrate our approach by designing two-dimensional elastic metamaterials that realize Veselago lenses, zero-dispersion bands and topological surface phonons. While our selected examples are within the mechanical domain, the same design principle can be applied to acoustic, thermal and photonic metamaterials composed of weakly interacting unit cells.

识别导致具有所需功能的超材料的材料几何形状对本领域提出了挑战。离散或降阶模型提供了对复杂现象的简要描述，例如负折射或拓扑表面状态。因此，将几何构件组合以复制呈现所需特征的离散模型代表了一种有前途的方法。但是，没有可靠的方法来解决这样的逆问题。在这里，我们介绍“微扰超材料”，这是由弱相互作用的晶胞组成的一类超材料。弱相互作用使我们能够将离散模型的每个元素与超材料的各个几何特征相关联，从而实现系统的设计过程。我们通过设计实现Veselago透镜，零色散带和拓扑表面声子的二维弹性超材料来证明我们的方法。虽然我们选择的示例在机械领域内，但是相同的设计原理可以应用于由弱相互作用的晶胞组成的声学，热学和光子超材料。