On the Fourier representation of elastic immersed boundaries

This paper presents an efficient treatment of fluid/elastic-structure interactions that takes advantage of the Fourier representation of immersed boundaries. We assume that the fluid is incompressible with uniform density and viscosity and that the immersed boundaries have fixed topologies. These elastic bodies can have large deformations and evolve anywhere within the fluid domain. They may be thick and are assumed to be piecewise smooth. We process the fluid-structure coupling with the immersed boundary method of C. S. Peskin. We can take advantage of the Fourier representation of the immersed bodies in many ways. First, the use of Fourier expansions allows us to filter out the high frequencies of the spatial oscillations along the boundary vectors. Second, we can work with a smaller number of boundary points to represent the interface, while preserving the same level of accuracy as long as enough points are used in the force spreading process. Finally, the harmonic information gathered by the Fourier coefficients is useful to control some global properties of the immersed boundaries. For example, we introduce a technique that corrects the volume conservation issue of closed immersed boundaries by performing constrained optimization in the Fourier space. We illustrate our method with two applications: one is a suspension flow with a large number of elastic 'bubbles', the other is an interesting case of artificial motion based on inertia rather than on flapping fins or flagella.