Control of flow behavior in complex fluids using automatic differentiation

Abstract

Inverse design of complex flows is notoriously challenging because of the high cost of high dimensional optimization. Usually, optimization problems are either restricted to few control parameters, or adjoint-based approaches are used to convert the optimization problem into a boundary value problem. Here, we show that the recent advances in automatic differentiation (AD) provide a generic platform for solving inverse problems in complex fluids. To demonstrate the versatility of the approach, we solve an array of optimization problems related to active matter motion in Newtonian fluids, dispersion in structured porous media, and mixing in journal bearing. Each of these problems highlights the advantages of AD in ease of implementation and computational efficiency to solve high-dimensional optimization problems involving particle-laden flows.

Figures (6)

• Figure 1: Flowchart of the optimization algorithm using a differentiable Immersed Boundary solver. The immersed boundary trajectory is unrolled with a computation graph. $F_i$, represent the loss function, equality and inequality constraints.
• Figure 2: Optimal Porous Media for flow rate a) Sketch of the periodic porous medium. We optimize over the position of the central (dark blue) particle, the ratio of sphere radii ($\lambda$), and the aspect ratio $\xi$. b) The optimization trajectory of the loss function. The pictures show how the network topology changes at chosen optimization iterations. c) The value of control parameters during the optimization iterations. After about 50 iterations, each parameter converges to an optima
• Figure 3: Optimal Porous Media for Dispersion Optimization trajectory of the scaled dispersion coefficient ($D^*/D_m$) for tracer particles in the flow. We initialize the blue (orange) particles in the top (bottom) half of the gap at $x=0$. The figure shows porous medium geometry and the positions of two colors of tracer particles after 100,000 time steps at different optimization iterations. The dashed blue line represents the scaled dispersion coefficient for the simple square array ($\lambda=0$).
• Figure 4: Chaotic mixing in a journal bearing (a) Mixing problem set-up illustration, with two non-concentric cylinders whose centers are displaced by $d$ of different radii $R_1,R_2$ being rotated with different angular velocities $\dot\theta_{1,2}$. The optimization parameters not only include geometrical parameters but also Fourier coefficients of the forcing (5 for each cylinder). (b) Trajectory of the design parameters after a number of optimization iterations. After a small number of iterations they settle to an optimum (c) The transient profile of the mixing indicator. The blue curve corresponds to the case of defined parameters while the red curve corresponds to the case using optimized parameters. The pictures illustrate the mixing at various points of time. The inset figure shows the optimum rotation velocity of the outer and inner cylinders. Here, $T=\tau f$ where $f$ is the rotation frequency.
• Figure 5: Optimal Swimmer (a) Sketch of the swimming problem, where a swimmer with an elliptical cross section moves with velocity $U$, using a stroke $h(t)$ and $\theta(t)$. Both $h(t),\theta(t)$ are parameterized with $20$ parameters. (b) Convergence of these design parameters over 80 iterations for the case $Re=1000$. (c) Loss function (swimming efficiency, the ratio of generated thrust to power input) as a function of iteration. The efficiency dropped more than twofold during the optimization