The Immersed Boundary (IB) method is a mathematical framework forconstructing robust numerical methods to study fluid-structure interaction inproblems involving an elastic structure immersed in a viscous fluid. The IBformulation uses an Eulerian representation of the fluid and a Lagrangianrepresentation of the structure. The Lagrangian and Eulerian frames are coupledby integral transforms with delta function kernels. The discretized IBequations use approximations to these transforms with regularized deltafunction kernels to interpolate the fluid velocity to the structure, and tospread structural forces to the fluid. It is well-known that the conventionalIB method can suffer from poor volume conservation since the interpolatedLagrangian velocity field is not generally divergence-free, and so this cancause spurious volume changes. In practice, the lack of volume conservation isespecially pronounced for cases where there are large pressure differencesacross thin structural boundaries. The aim of this paper is to greatly reducethe volume error of the IB method by introducing velocity-interpolation andforce-spreading schemes with the properties that the interpolated velocityfield in which the structure moves is at least C1 and satisfies a continuousdivergence-free condition, and that the force-spreading operator is the adjointof the velocity-interpolation operator. We confirm through numericalexperiments in two and three spatial dimensions that this new IB method is ableto achieve substantial improvement in volume conservation compared to otherexisting IB methods, at the expense of a modest increase in the computationalcost. Further, the new method provides smoother Lagrangian forces (tractions)than traditional IB methods. The method presented here is restricted toperiodic computational domains. Its generalization to non-periodic domains isimportant future work.
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