Topology Optimization of Adaptive Structures: New Limits of Material Economy

Adaptive structures can counteract the effect of external loads and other environmental actions through active manipulation of the internal force flow (i.e., the load path) and geometry (i.e., form or shape). Previous studies have shown that adaptation enables significant mass and energy saving compared to conventional structures that resist the effect of loading solely through material strength and stiffness (i.e., passive structures). The computational synthesis of adaptive structures is a challenging process since it involves optimization of the structural layout as well as sensor and actuator placement, which is, generally, a Mixed-Integer Non-linear Programming (MINLP) problem. Previous formulations employ sizing and/or geometry optimization in combination with actuator placement optimization. No method has yet been formulated for the simultaneous synthesis of the structural topology, element sizing, and placement of actuators. This paper offers the first-ever formulation for the All-In-One (AIO) topology optimization of adaptive structures based on the Ground Structure approach. The objective function comprises the mass of structural elements and actuators. The design variables are structural topology, element cross-section areas, and actuator locations. State variables include element forces and deformations, nodal displacements, and actuator commands. Constraints ensure that feasible solutions satisfy equilibrium and geometric compatibility as well as limits on stress, stability, nodal displacements, and actuator forces. Auxiliary constraints are implemented to enable the simultaneous synthesis of the structural layout and actuator placement and linearize the formulation into a Mixed-Integer Linear Problem (MILP) that can be solved to a global optimum. Due to a large number of variables, the AIO formulation can be typically applied to small-scale problems. To reduce the computational cost, a two-step sequential formulation is developed and benchmarked against the AIO method. Results show that the sequential method produces solutions of similar quality compared to the AIO one albeit with a significantly reduced computational cost. Results confirm that the optimal adaptive solutions vastly outperform topology-optimized conventional (i.e., passive) solutions. Adaptive solutions approach the limit of material economy (fully stressed design, e.g., Michell trusses) and, in parallel, satisfy important constraints including displacements and stability that would not be possible without adaptation.