1
|
- *work supported in part by NIH
|
2
|
- Introduction and Motivation
- The Immersed Boundary method
- Initial Modeling of Simplified Models of Phonation
- Preliminary Results of Some 2D Simulations
- Developmental Issues
|
3
|
- Phonation involves coupled interaction and self-oscillation of vocal
folds-air system
- Many models to date treat
- Material properties’ dynamics with an approximation to the
aerodynamics
- Aerodynamics with fixed or externally prescribed motions of the folds
- Aerodynamics with material dynamics modeled using finite elements
|
4
|
- Goal of present studies: Devise
models of phonation in which vocal folds and the aerodynamics are
treated as the closely coupled systems they are, exhibiting
self-oscillation and physical volume flows
- Present approach makes use of the Immersed Boundary (IB) method
originally developed by Charles Peskin (Courant Institute) for the
mammalian heart
- Describe initial research into feasibility of using IB applied to models
of phonation
- Present some preliminary results & assess potential for elaboration
and development of IB based models of phonation
|
5
|
|
6
|
- Equations summarizing the Immersed Boundary method:
|
7
|
- Eulerian mesh with immersed boundary
|
8
|
- Given Xni evaluate force Fni
on boundary marker i due to other material points for all i=1, Nparticles in material (Lagrangian).
- Spread Fni to
Eulerian grid using Delta-function [for each layer if multiple layers].
Gives fn, the force on the fluid due to the boundary/material
points.
- Solve forced Navier-Stokes equations to update velocity field to time
level n+1, thus giving un+1
- at all Eulerian grid points.
- Gather Eulerian velocity field values surrounding a given particle using
Delta-function. Gives Un+1i .
- Use No-Slip condition to push boundary points to new values, so Xn+1i emerges.
|
9
|
- Delta-function weighted spread of force to some eulerian mesh points
|
10
|
- Interpolate Eulerian velocities to particle
|
11
|
- 2D Model Feasibility Studies
- Single layer model
- Properties of folds’ materials lumped into force constants of the
particles which make up the layer
- Choose force constants which promote self-oscillation
|
12
|
- Navier Stokes solver based on FFT and assumes periodic boundary
conditions.
- N-S solver used 256 x 512 Eulerian mesh
- Used 5332 particles to model the 2D vocal folds with distance between
particles = h/4
- IB method computes velocity field, pressure field, and the motion of the
model folds as one coupled system
- Here as an illustration we render velocity field and vorticity and
marker motion as (animated gif) movies
|
13
|
|
14
|
|
15
|
|
16
|
|
17
|
|
18
|
- Strengths:
- Air-Folds system treated as
intrinsically coupled system
- Uses Eulerian cartesian mesh for NS and Lagrangian mesh for material
- Easy to craft material properties—so modeling pathologies natural
- 2D model shows promise as it exhibits many of the necessary properties
- Weaknesses:
- Currently for inompressible systems
- Due to computational time constraints not yet doing air, only
water—factor of 70 longer time needed for air—under development
|
19
|
- IB method has promise as a potentially viable tool for modeling
phonation
- 2D model exhibits self-oscillation and flow properties which indicate
promise for the IB approach
- Further developments underway
- Further work on modeling folds material properties
- Air as fluid simulations are began—considerable computational
constraints on sequential code imply probable need to move to parallel
IB method
|
20
|
- Add multiple layers in 2D, optimize material properties to obtain
oscillations in physical ranges (already under development)
- Single layer model in 3D
- Multiple layer models in 3D
- Parallel version of method in 2D (using MPI)
- Develop compressible immersed boundary method
- Coupling to model vocal tracts
|
21
|
|