Topology optimization of passively moving rigid bodies in unsteady flows

TL;DR

The paper presents a topology-optimization framework for rigid bodies that move passively under unsteady fluid flows, coupling fluid dynamics and rigid-body motion in a time-dependent, two-way manner. It leverages a grid-separation approach with the Lattice Kinetic Scheme (LKS) to handle unsteady flows and uses an adjoint-based sensitivity analysis (ALKS) to drive gradient-based optimization via MMA, augmented by a density filter and a continuation scheme. The method is demonstrated on 2D and 3D problems, including a 2D sail, a 2D turbine, and a 3D turbine, showing airfoil-like and turbine-like optimized shapes and highlighting the effects of grayscale regions and mesh resolution on binarization outcomes. This approach enables efficient design of passive locomotion devices such as sails and turbines in unsteady environments, with clear pathways for extensions to finer meshes and turbulent regimes.

Abstract

This study proposes the topology optimization method for moving rigid bodies subjected to forces from fluid flow, such as sails and turbines, with an unsteady time-dependent formulation. Unlike existing topology optimization frameworks in which rigid-body motion drives the flow, which is referred to as $\textit{active}$, the present study considers rigid-body motion induced by fluid forces, i.e., $\textit{passive}$. The equations of motion governing the rigid-body dynamics are solved in a coupled manner with the continuity equation and the momentum conservation equations. The rigid body is represented on a design grid that is separated from the analysis grid on which the state and adjoint fields are defined. After updating the rigid body motion, the body is mapped onto the analysis grid. The fluid equations are solved using the lattice kinetic scheme, an extended version of the lattice Boltzmann method, owing to its suitability for unsteady flows. Design sensitivities based on the adjoint variable method are presented and applied to two- and three-dimensional problems involving translational and rotational motions. The optimized shapes for each problem are discussed from a physical perspective and compared with a reference shape or their binarized counterparts, providing insights into the effectiveness of the proposed method as well as its limitations.

Figures (39)

• Figure 1: Relationship between the design and analysis domains
• Figure 2: D2Q9
• Figure 3: D3Q15
• Figure 5: Optimization procedure
• Figure 6: Design setting