Topology optimization is a powerful tool for designing structures in many fields, but has been limited to static or passively moving objects made of hard materials. Designing soft and actively moving objects, such as soft robots equipped with actuators, poses a challenge because the optimal structure depends on how the object will move and simulating dynamics problems is difficult. We propose "4D topology optimization," which extends density-based topology optimization to include time, as a method for optimizing the structure and movement of self-actuating soft bodies. The method represents the layout of both material and time-varying actuation using multi-index density variables distributed over the design domain, thus allowing simultaneous optimization of structure and movement using gradient-based methods. Forward and backward simulations of soft bodies are done using the material point method, a Lagrangian--Eulerian hybrid approach, implemented on a recent automatic differentiation framework. We present several numerical examples of designing self-actuating soft bodies targeted for locomotion, posture control, and rotation tasks. The results demonstrate that our method can successfully design complex shaped and biomimetic moving soft bodies due to its high degree of design freedom.

翻译：拓扑优化是众多领域中设计结构的强大工具，但此前一直局限于硬质材料制成的静态或被动移动物体。设计软质且主动移动的物体（例如配备驱动器的软体机器人）面临挑战，因为最优结构取决于物体运动方式，且动力学问题的模拟存在难度。我们提出"4D拓扑优化"——将基于密度的拓扑优化扩展至包含时间维度——作为优化自驱动软体结构及其运动的方法。该方法通过分布于设计域的多指标密度变量表征材料布局与时变驱动，从而支持基于梯度的梯度算法同步优化结构与运动。软体的正向与逆向仿真依托近期自动微分框架，采用拉格朗日-欧拉混合方法中的物质点法实现。我们给出了针对运动、姿态控制及旋转任务设计的自驱动软体的数值算例。结果表明，该方法凭借高度设计自由度，能够成功设计出复杂形态及仿生运动的软体结构。