Simulating an Elastic Ring with Bend and Twist by an Adaptive Generalized Immersed Boundary Method (Communications in Computational Physics, 2020)
Many problems involving the interaction of an elastic structure and a viscous fluid can be solved by the immersed boundary (IB) method. In the IB approachto such problems, the elastic forces generated by the immersed structure are applied tothe surrounding fluid, and the motion of the immersed structure is determined by thelocal motion of the fluid. Recently, the IB method has been extended to treat more general elasticity models that include both positional and rotational degrees of freedom.For such models, force and torque must both be applied to the fluid. The positionaldegrees of freedom of the immersed structure move according to the local linear velocity of the fluid, whereas the rotational degrees of freedom move according to the localangular velocity. This paper introduces a spatially adaptive, formally second-order accurate version of this generalized immersed boundary method. We use this adaptivescheme to simulate the dynamics of an elastic ring immersed in fluid. To describe theelasticity of the ring, we use an unconstrained version of Kirchhoff rod theory. Wedemonstrate empirically that our numerical scheme yields essentially second-orderconvergence rates when applied to such problems. We also study dynamical instabilities of such fluid-structure systems, and we compare numerical results produced byour method to classical analytic results from elastic rod theory.