Mean field control of droplet dynamics with high order finite element computations
CoRR(2024)
Abstract
Liquid droplet dynamics are widely used in biological and engineering
applications, which contain complex interfacial instabilities and pattern
formulation such as droplet merging, splitting, and transport. This paper
studies a class of mean field control formulation towards these droplet
dynamics. They are used to control and maintain the manipulation of 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. We lastly
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.
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