1	The Application of the Immersed Boundary Method to Modeling Phonation* Comer Duncan¹, Guangnian Zhai¹, and Ronald Scherer² (1) Department of Physics and Astronomy, (2) Department of Communication Disorders Bowling Green State University, Bowling Green OH 43403 *work supported in part by NIH
2	Immersed Boundary method for phonation-preliminary studies Introduction and Motivation The Immersed Boundary method Initial Modeling of Simplified Models of Phonation Preliminary Results of Some 2D Simulations Developmental Issues
3	Introduction and Motivation 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	Introduction and Motivation (cont’d) 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	Example from Peskin-McQueen work on the heart
6	The Immersed Boundary method: basic ideas Equations summarizing the Immersed Boundary method:
7	"Eulerian mesh with immersed boundary" Eulerian mesh with immersed boundary
8	Single Time Step in IB Method Given Xⁿ_ievaluate force Fⁿ_i on boundary marker i due to other material points for all i=1, N_particlesin material (Lagrangian). Spread Fⁿ_ito 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⁺¹_i. Use No-Slip condition to push boundary points to new values, so Xⁿ⁺¹_i emerges.
9	Force on particle spread to surrounding Eulerian mesh via delta-function Delta-function weighted spread of force to some eulerian mesh points
10	No-slip condition to obtain particle velocities from surrounding Eulerian mesh Interpolate Eulerian velocities to particle
11	Initial Modeling of Simplified Models of Phonation 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	2D Model (cont’d) 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	Vx Movie
14	Vy Movie
15	Vorticity
16	Markers and Model Folds Motion
17	Volume Flow for 2D model
18	Stengths and Weaknesses of IB method for Phonation 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	Conclusions 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	Developmental Issues 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	Example of Preliminary Multiple Layers System