Mathematical Modeling of Fluid-Structure Interactions

The mathematical modeling of fluid-structure interaction in the upper airways will be based on a simplified 2D model to study basic biomechanical mechanisms in OSAS. We will couple the compressible flow field in the pharynx to a model for the deformable wall in a two-way explicit coupling.

Our fluid solver builds upon a solid mathematical framework of high-order summation by Parts (SBP) difference operators and the simultaneous approximation term (SAT) treatment of boundary conditions [13]. This guarantees stable solutions and low dispersion errors which will be decisive for the accuracy of the simulated results.

We will employ a multi-block approach to represent the geometry in the upper airways. This allows using structured grids even for complex geometries. The deformation of the soft tissue in the palate will first be modeled in close collaboration with WP2 as a simple lumped-mass model and then extended to a more advanced nonlinear model allowing for large displacement. Due to the large deformations, an immersed boundary treatment will be preferred [14]. To reduce the complexity of our model, the unresolved parts such as the nose, mouth and lower airways will be treated as effective conductances in a 2D-1D coupling, for which the cross-sectional areas measured in WP1 will be used. Structural properties of the soft palate and the throat will be provided by WP2.

A high order explicit time integration scheme will be used for the sake of accuracy and easy parallelization. Our algorithm exhibits a large degree of fine-grained parallelism and can readily be implemented for solution on graphics processing units (GPUs) for faster computation. Powerful parallel computers provided by NTNU and the Norwegian Metacenter for Computational Science (NOTUR) will be employed for very large simulations (tens of breathing cycles) on fine grids. OpenMP has been used in our current structured grid codes.

The PhD candidate from IVT will be responsible for extending the formalism employed in the high-order method to the fully coupled fluid-structure system and implementing the mathematical model. Parameter studies on fixed geometries will also be valuable in assessing the basic mechanisms of OSAS. Similar models have recently been used to model fluid-structure interactions in human phonation in the larynx [13]. The resulting tool for FSI in the upper airways will be more accurate than available commercial software. Moreover, it will take into account acoustic and thermal effects, which are commonly neglected. The improved accuracy provided by the WP3 FSI model will be used as a calibration standard for the CFD model being developed in WP4.