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Modeling Elasstic Shells Immersed in Fluid

E. Givelberg

TL;DR

The paper develops a practical method to simulate elastic shells immersed in viscous incompressible fluids by extending the immersed boundary method with shell theory derived from differential geometry under the Kirchhoff-Love and plane-stress hypotheses. It provides a complete discretization pipeline, including geometry-dependent shell forces and a coupled fluid–shell time-stepping scheme, and demonstrates convergence on a basilar-membrane–like model. A key result is the observation of a traveling wave from base to apex in response to an external fluid impulse, aligning with cochlear mechanics. This work lays the groundwork for three-dimensional cochlear simulations and other fluid–shell interaction problems, albeit with high computational demands for full-scale models.

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.

Modeling Elasstic Shells Immersed in Fluid

TL;DR

The paper develops a practical method to simulate elastic shells immersed in viscous incompressible fluids by extending the immersed boundary method with shell theory derived from differential geometry under the Kirchhoff-Love and plane-stress hypotheses. It provides a complete discretization pipeline, including geometry-dependent shell forces and a coupled fluid–shell time-stepping scheme, and demonstrates convergence on a basilar-membrane–like model. A key result is the observation of a traveling wave from base to apex in response to an external fluid impulse, aligning with cochlear mechanics. This work lays the groundwork for three-dimensional cochlear simulations and other fluid–shell interaction problems, albeit with high computational demands for full-scale models.

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.

Paper Structure

This paper contains 17 sections, 96 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Linear Elastic Deformations of Vector Bundles (Shell Theory)
  3. The Geometry of the Vector Bundle
  4. The Infinitesimal Deformation of an Elastic Bundle
  5. The Strain Tensor
  6. The Plane Stress Assumption.
  7. Variation of the Strain Energy
  8. The Plate Equation
  9. The Immersed Boundary Method
  10. The Equations of the Model
  11. The Numerical Method
  12. Computation of the Elastic Force
  13. An Application: Modeling a Shell Immersed in Fluid
  14. The Model Shell
  15. The Numerical Experiments
  16. ...and 2 more sections

Figures (4)

  • Figure 1: The thickness of the model shell.
  • Figure 2: The relative difference between ${\bf X}_{1,128}$ and ${\bf X}_{2,128}$.
  • Figure 3: The relative difference between ${\bf X}_{0.5,128}$ and ${\bf X}_{0.5,64}$.
  • Figure 4: The traveling wave in the test shell.