An Immersed Boundary method with divergence-free velocity interpolation and force spreading

Abstract

The Immersed Boundary (IB) method is a mathematical framework for constructing robust numerical methods to study fluid–structure interaction in problems involving an elastic structure immersed in a viscous fluid. The IB formulation uses an Eulerian representation of the fluid and a Lagrangian representation of the structure. The Lagrangian and Eulerian frames are coupled by integral transforms with delta function kernels. The discretized IB equations use approximations to these transforms with regularized delta function kernels to interpolate the fluid velocity to the structure, and to spread structural forces to the fluid. It is well-known that the conventional IB method can suffer from poor volume conservation since the interpolated Lagrangian velocity field is not generally divergence-free, and so this can cause spurious volume changes. In practice, the lack of volume conservation is especially pronounced for cases where there are large pressure differences across thin structural boundaries. The aim of this paper is to greatly reduce the volume error of the IB method by introducing velocity-interpolation and force-spreading schemes with the properties that the interpolated velocity field in which the structure moves is at least scriptC1 and satisfies a continuous divergence-free condition, and that the force-spreading operator is the adjoint of the velocity-interpolation operator. We confirm through numerical experiments in two and three spatial dimensions that this new IB method is able to achieve substantial improvement in volume conservation compared to other existing IB methods, at the expense of a modest increase in the computational cost. Further, the new method provides smoother Lagrangian forces (tractions) than traditional IB methods. The method presented here is restricted to periodic computational domains. Its generalization to non-periodic domains is important future work.

A Gaussian-like immersed-boundary kernel with three continuous derivatives and improved translational invariance

“…existing IB kernels by their nonzero second-moment constant K. A special choice of K leads to a 5-point or 6-point IB kernel that is C 3 and features substantially improved translational invariance compared with the existing standard IB kernels. Recently, we have successfully applied the new IB kernels to a new IB method with an exactly divergence-free interpolated velocity field [9], in which derivatives of the discrete delta function are involved, and have achieved 10 3 to 10 5 times improvements in volume conservation of the IB method. We believe that, in many other applications in which derivatives of the IB kernel are needed, the improved grid invariance and regularity of the new IB kernels will be worth its extra computational cost.…”

Numerical methods for immersed FSI with thin-walled structures

“…While the impact of the former can be mitigated via mesh refinement, the latter is known to yield major mass conservation issues across the interface. It is worth noting that similar poor mass conservation was encountered with the original immersed boundary method using finite difference approximations in space (see, e.g., [41,5]). In the variational setting of Algorithm 2, the use of alternative spatial approximations, based on globally discontinuous pressures and higher order polynomials for the velocities, has been investigated in [4,8,7,10] with the purpose of overcoming this issue.…”

Leveraging collective effects in externally driven colloidal suspensions: experiments and simulations

“…The corresponding cost depends on the fluid solver and on the discretization of the particles. FH has been successfully implemented into various numerical methods for particulate flows with linear scaling (O(N )) such as the Immersed Boundary Method [145,146,147,148,149], Distributed Lagrange Multipliers [150], Fluid Particle Dynamics [151], Finite Element Method [152,153], Force Coupling Method [154,155], Boundary Element Method [156] or Particle Mesh Ewald Method [157,158].…”