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Mean field control of droplet dynamics with high order finite element computations

Guosheng Fu, Hangjie Ji, Will Pazner, Wuchen Li

TL;DR

This work develops a mean field control framework for droplet dynamics governed by a high-order thin-film (lubrication) equation with non-mass-conserving evaporation and active suspensions. By casting the dynamics as a gradient flow in a generalized Wasserstein-like metric with nonlinear mobilities, the authors formulate an MFC problem, derive linearized and auxiliary-variable reformulations, and obtain a saddle-point system suitable for a high-order space-time finite element discretization. A generalized PDHG solver is paired with high-order FEM, and a fully discrete JKO-type scheme is provided for time stepping, enabling numerical demonstrations of droplet transport, bead-up, spreading, merging, and splitting driven by an active-field control. The results illustrate the viability of mean field control to actuate complex interfacial dynamics and provide a computational framework that could inform experimental design and optimization in digital microfluidics and related areas.

Abstract

Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formation such as droplet merging, splitting, and transport. This paper studies a class of mean field control formulations for these droplet dynamics, which can be used to control and manipulate droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. As an example, a lubrication equation for a thin volatile liquid film laden with an active suspension is developed, with control achieved through its activity field. Lastly, we apply the primal-dual hybrid gradient algorithm with high-order finite element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism.

Mean field control of droplet dynamics with high order finite element computations

TL;DR

This work develops a mean field control framework for droplet dynamics governed by a high-order thin-film (lubrication) equation with non-mass-conserving evaporation and active suspensions. By casting the dynamics as a gradient flow in a generalized Wasserstein-like metric with nonlinear mobilities, the authors formulate an MFC problem, derive linearized and auxiliary-variable reformulations, and obtain a saddle-point system suitable for a high-order space-time finite element discretization. A generalized PDHG solver is paired with high-order FEM, and a fully discrete JKO-type scheme is provided for time stepping, enabling numerical demonstrations of droplet transport, bead-up, spreading, merging, and splitting driven by an active-field control. The results illustrate the viability of mean field control to actuate complex interfacial dynamics and provide a computational framework that could inform experimental design and optimization in digital microfluidics and related areas.

Abstract

Liquid droplet dynamics are widely used in biological and engineering applications, which contain complex interfacial instabilities and pattern formation such as droplet merging, splitting, and transport. This paper studies a class of mean field control formulations for these droplet dynamics, which can be used to control and manipulate droplets in applications. We first formulate the droplet dynamics as gradient flows of free energies in modified optimal transport metrics with nonlinear mobilities. We then design an optimal control problem for these gradient flows. As an example, a lubrication equation for a thin volatile liquid film laden with an active suspension is developed, with control achieved through its activity field. Lastly, we apply the primal-dual hybrid gradient algorithm with high-order finite element methods to simulate the proposed mean field control problems. Numerical examples, including droplet formation, bead-up/spreading, transport, and merging/splitting on a two-dimensional spatial domain, demonstrate the effectiveness of the proposed mean field control mechanism.
Paper Structure (19 sections, 1 theorem, 88 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 19 sections, 1 theorem, 88 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Mean field control of droplet dynamics
  4. Droplet dynamics induced distances and MFCs
  5. MFC reformulations
  6. Saddle-point problem
  7. High order discretizations and optimization algorithms
  8. The high-order finite element scheme
  9. A generalized PDHG algorithm
  10. A fully discrete JKO scheme to the PDE \ref{['eq:model']}
  11. Discussions
  12. Numerical results
  13. JKO scheme for the PDE \ref{['eq:model']}.
  14. One-dimensional example
  15. Two-dimensional example
  16. ...and 4 more sections

Key Result

Proposition 3.1

Let $(h, \bm{m}, s)$ be the critical point system of the MFC problem mfcR2. Then there exists a function $\phi\colon [0, T]\times\Omega\rightarrow \mathbb{R}$, such that and where $\mathcal{I}(h)$ is the generalized Fisher information functional given in fisher, such that with initial and terminal time conditions

Figures (10)

  • Figure 1: Schematic figure for the mean field control from the initial droplet profile $h_0(x,y)$ to the target droplet profile $h_{T}(x,y)$.
  • Figure 2: Snapshots of the surface height for 1D thin-film equation \ref{['eq:model']} at different times. Dashed black line: FEM scheme \ref{['FEM']} with time step size $\Delta t = 10^{-4}$; Dotted red line: Approximated JKO scheme \ref{['saddleH-JKOa']} with $\Delta t = 10^{-4}$ using the Newton-Raphson solver; Blue line: Approximated JKO scheme \ref{['saddleH-JKOa']} with $\Delta t = 10^{-3}$ using the Newton-Raphson solver; Green line: Approximated JKO scheme \ref{['saddleH-JKOa']} with $\Delta t = 10^{-3}$ using the PDHG solver with tolerance $tol=10^{-8}$; Magenta line: Approximated JKO scheme \ref{['saddleH-JKOa']} with $\Delta t = 10^{-3}$ using the PDHG solver with tolerance $tol=10^{-7}$.
  • Figure 3: Snapshots of the surface height contour for 2D thin-film equation \ref{['eq:model']} at different times. First row: numerical solutions for FEM \ref{['FEM']}, $\Delta t=0.0001$; Second row: numerical solutions for the approximated JKO scheme \ref{['saddleH-JKOa']} with the Newton-Raphson solver for \ref{['newton']} in each JKO step, $\Delta t=0.0001$; Third row: numerical solutions for the approximated JKO scheme \ref{['saddleH-JKOa']} with the Newton-Raphson solver for \ref{['newton']} in each JKO step, $\Delta t=0.001$; Last row: numerical solutions for the approximated JKO scheme \ref{['saddleH-JKOa']} with PDHG solver in Algorithm \ref{['alg:1']} for each JKO step, $\Delta t=0.001$, $tol = 10^{-8}$.
  • Figure 4: Case 1 (Droplet transport): Snapshots of (top row) 3D plots of the controlled surface height $h$, (middle row) contour plots of the controlled surface height, and (bottom row) contour plots of the activity field $\zeta$ at different times. The corresponding animation video can be found in the GitHub repository github.
  • Figure 5: Case 2 (Droplet spreading): Snapshots of (top row) 3D plots of the controlled surface height $h$, (middle row) contour plots of the controlled surface height, and (bottom row) contour plots of the activity field $\zeta$ at different times. The corresponding animation video can be found in the GitHub repository github.
  • ...and 5 more figures

Theorems & Definitions (16)

  • Example 1
  • Example 2
  • Example 3
  • Remark 4: Rescaling
  • Definition 1: Distance functional
  • Remark 2: On convexity
  • Definition 3: MFC for droplet dynamics
  • Remark 4: JKO temporal discretization to \ref{['eq:model']}
  • Definition 5: MFC reformulation I
  • Proposition 3.1: MFC systems of droplet dynamics
  • ...and 6 more