Fluid-structure interaction (FSI) is ubiquitous in nature and occurs at molecular to environmental scales, from the writhing of DNA in nucleoplasm, to the beating of cilia and flagella and the projection of lamellipodia and bleb-like protrusions by motile cells, to the flow of blood in the heart, to swimming fish and flying birds and insects, to the dispersal of seeds and pollen in the wind. This talk will describe numerical methods for FSI, focusing on a series of extensions of the immersed boundary (IB) method for fluid-structure interaction. Accurate numerical simulations of FSI require fine computational meshes, but in many cases, this requirement is spatially localized to flows near the immersed structures. For the flow away from the immersed structures, the need for high spatial resolution can be lessened. To improve the computational performance of the IB method in such situations, we have developed efficient versions of the IB method that use block-structured adaptive mesh refinement (AMR) that locally deploy high spatial resolution where it is most needed.

Different approaches are needed for FSI involving rigid and elastic structures, but both can be addressed within the framework of the IB method. We shall discuss IB methods for FSI with prescribed structural kinematics that exactly impose kinematic constraints using Lagrange multipliers as well as different penalty methods that offer efficient approximations to these exactly constrained formulations that are suitable for different classes of applications. We shall also discuss IB methods for flexible bodies that use nonlinear structural dynamics formulations that may be discretized using standard Lagrangian finite element methods. Finally, we shall describe extensions of these methods that aim to achieve higher-order convergence rates using ideas introduced within the context of the immersed interface methods. These methods are implemented in, and distributed via, the IBAMR software infrastructure. We also shall survey several modeling applications of these IB methods in biology and medicine, including flagellar mechanics, aquatic locomotion and neuro-mechanical feedback, cardiac fluid dynamics, and esophageal transport.