[Paper Review] Modeling Elasstic Shells Immersed in Fluid

Edward Givelberg|ArXiv.org|Jan 27, 2003

Lattice Boltzmann Simulation Studies10 references19 citations

TL;DR

This paper presents a numerical method based on the immersed boundary framework extended to elastic shells using Kirchhoff-Love and planar stress shell theories, with differential geometry underpinning the formulation. The method successfully simulates a traveling wave in a basilar membrane prototype, reproducing the cochlear phenomenon of wave propagation from base to apex, validating its use in full 3D cochlear modeling.

ABSTRACT

We describe a numerical method to simulate an elastic shell immersed in a viscous incompressible fluid. The method is developed as an extension of the immersed boundary method using shell equations based on the Kirchhoff-Love and the planar stress hypotheses. A detailed derivation of the shell equations used in the numerical method is presented. This derivation as well as the numerical method, use techniques of differential geometry in an essential way. Our main motivation for the development of this method is its use in the construction of a comprehensive three-dimensional computational model of the cochlea (the inner ear). The central object of study within the cochlea is the ``basilar membrane'', which is immersed in fluid and whose elastic properties rather resemble those of a shell. We apply the method to a specific example, which is a prototype of a piece of the basilar membrane and study the convergence of the method in this case. Some typical features of cochlear mechanics are already captured in this simple model. In particular, numerical experiments have shown a traveling wave propagating from the base to the apex of the model shell in response to external excitation in the fluid.

Motivation & Objective

• To develop a computational method for simulating elastic shells immersed in viscous incompressible fluids, motivated by the need to model the cochlea’s mechanical behavior.
• To extend the immersed boundary method to include shell dynamics using Kirchhoff-Love and planar stress hypotheses, grounded in differential geometry.
• To construct a foundation for a full 3D computational model of the cochlea, focusing on the basilar membrane’s role in auditory signal processing.
• To validate the method’s convergence and accuracy using a prototype basilar membrane model under impulsive fluid excitation.
• To reproduce key features of cochlear mechanics, such as base-to-apex traveling wave propagation, in a simplified numerical setup.

Proposed method

• The method extends the immersed boundary framework by incorporating shell equations derived from the Kirchhoff-Love and planar stress hypotheses, modeling the elastic shell’s in-plane and bending behavior.
• Differential geometry techniques are used to derive the shell equations, enabling accurate representation of curved, thin elastic structures in fluid-structure interaction.
• The fluid is governed by the incompressible Navier-Stokes equations, while the elastic shell is modeled as a network of force elements distributed along its mid-surface.
• A remapping procedure transfers forces between the Eulerian fluid grid and the Lagrangian shell nodes, ensuring conservation and consistency in the fluid-structure coupling.
• The time integration uses a fractional step method with a small time step constrained by the CFL condition and stiffness of the immersed boundaries.
• Convergence is analyzed by refining the fluid mesh and time step, showing first-order convergence when the time step is proportional to the fluid mesh width.

Experimental results

Research questions

• RQ1 Can the immersed boundary method be extended to model elastic shells with curvature and in-plane stresses in viscous fluid flow?
• RQ2 Does the resulting numerical method reproduce the experimentally observed base-to-apex traveling wave in the basilar membrane?
• RQ3 What is the convergence behavior of the method when the fluid mesh and time step are refined?
• RQ4 Can the method capture the mechanical response of a shell with spatially varying compliance under fluid excitation?
• RQ5 Is the method scalable and stable for large-scale simulations of the full 3D cochlear geometry?

Key findings

• The numerical method exhibits first-order convergence when the time step is reduced proportionally to the fluid mesh width, confirming stability and consistency.
• A traveling wave propagates from the base to the apex of the shell model in response to an impulsive fluid force, matching von Békésy’s experimental observations of cochlear wave mechanics.
• The wave propagation is driven by the spatial variation in shell compliance, with stiffer regions near the base initiating the wave and the wave moving toward more compliant apex regions.
• The method successfully captures the displacement dynamics of the shell, with visualizations showing downward fluid impulse followed by elastic rebound and wave propagation.
• The method was validated on a prototype basilar membrane model and later applied to construct a full 3D cochlear model, demonstrating feasibility despite high computational demands.
• The simulation required a time step of approximately 50 nanoseconds for a 100 μm fluid mesh, consistent with stability constraints from the CFL condition and boundary stiffness.