Folding can transform mundane objects such as napkins into stunning works of art. However, finding new folding transformations for sheet materials is a challenging problem that requires expertise and real-world experimentation. In this paper, we present Modal Folding -- an automated approach for discovering energetically optimal folding transformations, i.e., large deformations that require little mechanical work. For small deformations, minimizing internal energy for fixed displacement magnitudes leads to the well-known elastic eigenmodes. While linear modes provide promising directions for bending, they cannot capture the rotational motion required for folding. To overcome this limitation, we introduce strain-space modes -- nonlinear analogues of elastic eigenmodes that operate on per-element curvatures instead of vertices. Using strain-space modes to determine target curvatures for bending elements, we can generate complex nonlinear folding motions by simply minimizing the sheet's internal energy. Our modal folding approach offers a systematic and automated way to create complex designs. We demonstrate the effectiveness of our method with simulation results for a range of shapes and materials, and validate our designs with physical prototypes.

翻译：折叠能将餐巾等普通物品转化为令人惊叹的艺术品。然而，为片材发现新的折叠变换是一项需要专业知识和实体实验的挑战性问题。本文提出模态折叠——一种自动发现能量最优折叠变换（即所需机械功较小的大变形）的方法。在小变形条件下，针对固定位移幅值最小化内能会导出经典的弹性本征模态。尽管线性模态为弯曲提供了有前景的方向，但无法捕捉折叠所需的旋转运动。为克服这一局限，我们引入应变空间模式——弹性本征模态的非线性对应物，其作用于每个单元的曲率而非顶点。通过利用应变空间模式确定弯曲单元的目标曲率，仅需最小化片材内能即可生成复杂的非线性折叠运动。我们的模态折叠方法为创建设计复杂的结构提供了系统化自动途径。通过针对多种形状与材料的仿真结果验证了该方法的有效性，并利用物理原型验证了设计的可行性。