Design requirements for moving parts in mechanical assemblies are typically specified in terms of interactions with other parts. Some are purely kinematic (e.g., pairwise collision avoidance) while others depend on physics and material properties (e.g., deformation under loads). Kinematic design methods and physics-based shape/topology optimization (SO/TO) deal separately with these requirements. They rarely talk to each other as the former uses set algebra and group theory while the latter requires discretizing and solving differential equations. Hence, optimizing a moving part based on physics typically relies on either neglecting or pruning kinematic constraints in advance, e.g., by restricting the design domain to a collision-free space using an unsweep operation. In this paper, we show that TO can be used to co-design two or more parts in relative motion to simultaneously satisfy physics-based criteria and collision avoidance. We restrict our attention to maximizing linear-elastic stiffness while penalizing collision measures aggregated in time. We couple the TO loops for two parts in relative motion so that the evolution of each part's shape is accounted for when penalizing collision for the other part. The collision measures are computed by a correlation functional that can be discretized by left- and right-multiplying the shape design variables by a pre-computed matrix that depends solely on the motion. This decoupling is key to making the computations scalable for TO iterations. We demonstrate the effectiveness of the approach with 2D and 3D examples.

翻译：机械装配中运动部件的设计要求通常根据其与其他部件的相互作用来规定。其中一些要求是纯运动学的（例如成对碰撞避免），而另一些则依赖于物理和材料特性（例如载荷下的变形）。运动学设计方法与基于物理的形状/拓扑优化（SO/TO）分别处理这些要求。由于前者使用集合代数和群论，而后者需要离散化并求解微分方程，二者鲜有结合。因此，基于物理的运动部件优化通常依赖于预先忽略或修剪运动学约束，例如通过使用非扫掠操作将设计域限制在无碰撞空间内。本文证明，拓扑优化可用于协同设计两个或多个相对运动的部件，以同时满足基于物理的准则和碰撞避免要求。我们将研究重点限定在最大化线弹性刚度，同时对随时间累积的碰撞度量进行惩罚。通过耦合两个相对运动部件的拓扑优化循环，使得在惩罚一个部件的碰撞时能够考虑另一部件形状的演变。碰撞度量通过相关泛函计算，该泛函可通过将形状设计变量左乘和右乘一个仅取决于运动的预计算矩阵来离散化。这种解耦是实现拓扑优化迭代计算可扩展性的关键。我们通过二维和三维算例验证了该方法的有效性。