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Simulating the dynamics of discretized interacting structures whose relationship is dictated by a kernel function gives rise to a large dense matrix. We propose a multigrid solver for such a matrix that exploits not only its data-sparsity resulting from the decay of the kernel function but also the regularity of the geometry of the structures and the quantities of interest distributed on them. Like the well-known multigrid method for large sparse matrices arising from boundary-value problems, our method requires a smoother for removing high-frequency terms in solution errors, a strategy for coarsening a grid, and a pair of transfer operators for exchanging information between two grids. We develop new techniques for these processes that are tailored to a kernel function acting on discretized interacting structures. They are matrix-free in the sense that there is no need to construct the large dense matrix. Numerical experiments on a variety of bio-inspired microswimmers immersed in a Stokes flow demonstrate the effectiveness and efficiency of the proposed multigrid solver. In the case of free swimmers that must maintain force and torque balance, additional sparse rows and columns need to be appended to the dense matrix above. We develop a matrix-free fast solver for this bordered matrix as well, in which the multigrid method is a key component. 

In this work, we outline a methodology for determining optimal helical flagella placement and phase shift that maximize fluid pumping through a rectangular flow meter above a simulated bacterial carpet. This method uses a Genetic Algorithm (GA) combined with a gradient-based method, the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm, to solve the optimization problem and the Method of Regularized Stokeslets (MRS) to simulate the fluid flow. This method is able to produce placements and phase shifts for small carpets and could be adapted for implementation in larger carpets and various fluid tasks. Our results show that given identical helices, optimal pumping configurations are influenced by the size of the flow meter. We also show that intuitive designs, such as uniform placement, do not always lead to a high-performance carpet. 

Abstract We study steady-state thin films on chemically heterogeneous substrates of finite size, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1D steady-state solutions that exist on such substrates into six different branches and develop asymptotic estimates for the steady states on each branch. Using perturbation expansions, we show that leading-order solutions provide good predictions of the steady-state thin films on stepwise-patterned substrates. We show how the analysis in one dimension can be extended to axisymmetric solutions. We also examine the influence of the wettability contrast of the substrate pattern on the linear stability of droplets and the time evolution for dewetting on small domains. Results are also applied to describe 2D droplets on hydrophilic square patches and striped regions used in microfluidic applications.