Topology optimization is a powerful tool utilized in various fields for structural design. However, its application has primarily been restricted to static or passively moving objects, mainly focusing on hard materials with limited deformations and contact capabilities. Designing soft and actively moving objects, such as soft robots equipped with actuators, poses challenges due to simulating dynamics problems involving large deformations and intricate contact interactions. Moreover, the optimal structure depends on the object's motion, necessitating a simultaneous design approach. To address these challenges, we propose "4D topology optimization," an extension of density-based topology optimization that incorporates the time dimension. This enables the simultaneous optimization of both the structure and self-actuation of soft bodies for specific dynamic tasks. Our method utilizes multi-indexed and hierarchized density variables distributed over the spatiotemporal design domain, representing the material layout, actuator layout, and time-varying actuation. These variables are efficiently optimized 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 self-actuating soft body designs aimed at achieving locomotion, posture control, and rotation tasks. The results demonstrate the effectiveness of our method in successfully designing soft bodies with complex structures and biomimetic movements, benefiting from its high degree of design freedom.

翻译：拓扑优化是广泛应用于各领域结构设计的有力工具。然而，其应用迄今主要局限于静态或被动运动物体，且主要关注变形能力与接触能力有限的硬质材料。设计配备驱动器的软机器人等具备主动运动能力的柔性物体极具挑战性，因其需处理涉及大变形与复杂接触相互作用的动力学问题。此外，最优结构取决于物体运动方式，因此需采用同步设计方法。为应对这些挑战，我们提出"4D拓扑优化"——一种融合时间维度的基于密度拓扑优化的扩展方法，旨在针对特定动态任务同步优化软体的结构与自驱机制。该方法在时空设计域上分布多索引分层密度变量，分别表征材料布局、驱动器布局与时变驱动行为，并通过基于梯度的优化方法高效求解。基于近期自动微分框架，我们采用物质点法（一种拉格朗日-欧拉混合方法）实现软体的正向与反向仿真。针对运动控制、姿态调整及旋转任务，本文展示了多个自驱软体设计的数值算例。结果表明，得益于其高度设计自由度，该方法能有效设计兼具复杂结构与仿生运动特性的软体系统。