Topology optimization of passively moving rigid bodies in unsteady flows

This paper proposes a topology optimization method for designing passively moving rigid bodies in unsteady fluid flows by coupling rigid-body dynamics with lattice kinetic fluid equations and applying an adjoint-based sensitivity analysis to optimize shapes for translational and rotational motions.

The Gist

Imagine you are trying to design the perfect sail for a boat or the perfect blade for a wind turbine. Usually, engineers start with a shape, put it in the wind, see how it moves, tweak the shape, and try again. This paper introduces a super-smart, automated way to do that, but with a twist: instead of the shape just sitting there while wind blows over it, the shape is allowed to move because the wind is pushing it.

Here is a breakdown of how this "magic" works, using simple analogies:

1. The Big Idea: The "Passive" Dancer

Most computer programs that design shapes assume the object is glued to the floor (like a bridge or a stationary pipe). If you want to design a moving part, like a fan blade, you usually tell the computer, "Spin this blade at 100 rotations per minute," and the computer figures out the airflow.

This paper flips the script. It treats the object like a dancer on a dance floor.

• The Old Way (Active): You tell the dancer exactly how to move, and you watch how the air moves around them.
• The New Way (Passive): You don't tell the dancer how to move. You just set the music (the wind) playing, and you ask the computer to design the dancer's body so that the music naturally pushes them to spin or glide as far as possible. The dancer's movement is a result of the wind, not a command.

2. The Two-Grid Trick: The Map vs. The Terrain

To make this work, the computer uses a clever trick called "separated grids." Imagine you are drawing a map of a moving island on a piece of graph paper.

• The Design Grid (The Map): This is where the shape is drawn. It's like a sketchpad. The computer decides where the "solid" material (the island) and the "empty" space (the water) are here.
• The Analysis Grid (The Terrain): This is where the physics happens. It's a fixed grid of water and wind.

Every tiny fraction of a second, the "Map" (the shape) physically moves and rotates. Then, the computer projects that moving map onto the fixed "Terrain" grid to calculate how the wind pushes against it. After the wind pushes, the computer calculates how the object should move next, updates the Map, and repeats the cycle. It's like taking a photo of a moving object, calculating the wind force, moving the object, and taking the next photo instantly.

3. The "Ghost" Force (Brinkman Force)

How does the computer know where the solid object is? It uses a concept called the Brinkman force.
Think of the design area as a room filled with invisible, sticky honey.

• Where there is solid material, the honey is super thick and sticky. The wind can't move through it; it just pushes against the surface.
• Where there is empty space, the honey is thin or non-existent, so the wind flows freely.
  The computer doesn't need to draw a hard line; it just adjusts the "stickiness" of the honey at every point. If the stickiness is high, it's a wall; if it's low, it's air. This allows the shape to morph smoothly from one form to another.

4. The "Time-Travel" Math (Adjoint Method)

To find the perfect shape, the computer has to know: "If I change this tiny dot of material here, how much better will the object move?"
Calculating this for every single dot would take forever. So, the authors use a method called the Adjoint Variable Method.

• The Analogy: Imagine you are trying to find the best path up a mountain in the dark. Instead of walking every possible path forward to see which is best, you shine a flashlight backward from the summit. The light shows you exactly which steps lead uphill most efficiently.
• In this paper, the "flashlight" runs backward through time, calculating how the wind forces and the object's motion would have reacted to every tiny change in the shape. This gives the computer a "sensitivity map" telling it exactly where to add or remove material to get the best result.

5. The Results: What Did They Build?

The team tested this on three scenarios:

1. The 2D Sail: They designed a shape that starts still and gets pushed by the wind to slide horizontally. The result looked like a curved airplane wing (airfoil). The wind pushed harder on the top than the bottom, creating lift that pulled the object forward.
2. The 2D Turbine: They designed a shape that spins. The result looked like a four-bladed propeller. The wind hit the curved blades, creating a twist that made it rotate.
3. The 3D Turbine: They did the same thing in 3D. The result looked like a real-world wind turbine.

6. The "Grayscale" Issue

In these computer designs, the edges of the shapes aren't always perfectly sharp black-and-white lines. Sometimes they are "grayscale"—a bit of both solid and air.

• In the 2D examples, the authors found that even if they made the shape perfectly sharp (black and white), the performance was almost the same. The "fuzzy" edges didn't hurt the result.
• In the 3D example, the "fuzzy" edges mattered more. Because the computer grid was a bit "chunky" (low resolution), the fuzzy edges changed how the wind hit the blades. This suggests that for complex 3D shapes, we need a finer "map" to get a perfect result.

Summary

This paper presents a new way to design moving machines (like sails or turbines) where the computer figures out the shape and the movement simultaneously. It treats the object like a passive dancer pushed by the wind, uses a "sticky honey" trick to define the shape, and runs a backward-time math simulation to find the most efficient form. The result is shapes that naturally look like wings and propellers, optimized to move as far or spin as fast as possible under fluid forces.

Technical Summary

Technical Summary: Topology Optimization of Passively Moving Rigid Bodies in Unsteady Flows

Problem Statement
This study addresses the topology optimization of rigid bodies that are driven by fluid-flow-induced forces, such as sails and turbines. Unlike traditional topology optimization in fluid dynamics where rigid-body motion is prescribed (active) to drive the flow, or where particles are tracked in a one-way coupling where they do not affect the fluid, this work focuses on a two-way coupled system. Here, the rigid body's motion is passive, meaning its trajectory and rotation are entirely determined by the fluid forces acting upon it. The governing challenge involves solving the unsteady fluid flow equations simultaneously with the rigid-body equations of motion, where the fluid forces serve as inputs to the motion equations, and the body's position serves as a boundary condition for the fluid.

Methodology
The proposed framework integrates several numerical techniques to handle the complex interaction between the moving solid and the unsteady fluid:

• Governing Equations: The fluid flow is modeled using the incompressible Navier-Stokes equations, solved via the Lattice Kinetic Scheme (LKS), an extended version of the Lattice Boltzmann Method (LBM) suitable for unsteady flows. The rigid-body dynamics are governed by Newton-Euler equations, where the forces and torques are computed as the reaction of the Brinkman force within the fluid domain.
• Separated Design-Analysis Grids: A core component of the methodology is the separation of the design grid (where the shape is defined) from the analysis grid (where fluid state variables are computed). The design grid undergoes rigid-body motion at each time step and is mapped onto the analysis grid using a weight function. This allows for the direct representation of rigid-body motion without remeshing the fluid domain.
• Adjoint Sensitivity Analysis: To compute design sensitivities efficiently, the study employs the Adjoint Lattice Kinetic Scheme (ALKS). The authors derive continuous adjoint equations that couple the fluid-flow equations with the rigid-body equations of motion. This formulation treats the rigid-body motion variables (position, velocity, rotation angle, and angular velocity) as additional state fields within the adjoint system.
• Optimization Algorithm: The shape is updated using the Method of Moving Asymptotes (MMA). To ensure manufacturable, binary-like designs, a density filter with Heaviside projection is utilized. A continuation scheme is applied to the convexity parameter and the projection steepness parameter to guide the optimization from a diffuse initial state to a clear solid-fluid distribution.
• Periodic Approximation: For problems involving cyclic motion (e.g., turbines), a periodic problem approximation is employed. Initial values for state and adjoint fields at the start of an optimization iteration are set to the terminal values of the previous iteration, reducing computational cost by avoiding the simulation of transient startup phases for every iteration.

Key Contributions

1. Passive Motion Formulation: The paper presents a novel topology optimization framework specifically for rigid bodies driven by fluid forces, distinguishing it from active motion or one-way coupling particle tracking.
2. Coupled Adjoint Derivation: It provides a rigorous derivation of adjoint equations and design sensitivities that fully couple the fluid dynamics with the rigid-body equations of motion, treating the body's kinematic variables as state fields.
3. Grid Separation Implementation: The study demonstrates the effectiveness of the grid separation approach for unsteady, moving rigid body problems, facilitating the transfer of quantities like the Brinkman coefficient and solid velocity between moving and stationary grids.
4. Numerical Validation: The method is verified against finite difference approximations for both translational and rotational cases, confirming the accuracy of the derived sensitivities.

Results
The method was applied to three numerical examples:

• 2D Sail: The optimization maximized the horizontal displacement of a rigid body. The resulting shape resembled an airfoil, generating lift forces that propelled the body forward. The optimized shape significantly outperformed a reference shape with a flat upper surface.
• 2D Turbine: The objective was to maximize the rotation angle. By employing a cyclic periodic condition, the optimization produced a shape with four airfoil-like blades that rotated effectively under fluid forces. The results indicated that the presence of grayscale regions in the optimized shape had a negligible effect on performance when binarized.
• 3D Turbine: A three-dimensional rotating shape was optimized, yielding a turbine-like structure. However, due to computational limitations resulting in coarser mesh resolution, the binarized version showed non-negligible differences in phase and period compared to the optimized shape, highlighting the sensitivity of 3D thin structures to mesh resolution.

Significance and Claims
The authors claim that the proposed method effectively optimizes the shapes of devices like sails and turbines where performance is dictated by the interaction between fluid forces and rigid-body motion. The study demonstrates that unsteady analysis is necessary for such problems, as the effective angle of attack and forces change dynamically with the body's motion.

The paper modestly notes that while the method is effective, limitations exist, particularly in three-dimensional cases where mesh resolution constraints can lead to discrepancies between optimized and binarized shapes. The authors suggest that future work should focus on improving mesh resolution (e.g., via adaptive methods) and extending the framework to turbulent flows. The study concludes that the proposed coupled topology optimization approach provides a robust tool for designing fluid-driven moving components, offering insights into the physical mechanisms of lift and drag generation in unsteady regimes.