Simulation of nanoparticle transport in airways using Petrov-Galerkin finite element methods

Abstract

Nanoparticles with various diameters, i.e. 1 nm d d d 150 nm, were studied with respect to their transport and deposition properties in the human airways. A finite element code, written in C++, was developed that solved both the Navier-Stokes and Advection-Diffusion equation monolithically. When modeling nanoparticles, the regular finite element method becomes unstable, and, in order resolve this issue, various stabilization methods were consider including Streamline Upwind, Streamline Upwind Petrov-Galerkin and Galerkin Least Square. In order to validate the various types of stabilization, the stabilized finite element solution was compared to the analytical Graetz solution. The comparison was done by calculation an approximation of the L 2 - error, and the best stabilization method was found to be Galerkin Least Square. Also in this thesis, we found that the Crank-Nicolson time stepping scheme is not the best option for the human airways simulations problem, and this is due to both the complex nature of the geometry and the Crank-Nicolson method lacks the ability to damp out error when the problem is advection dominated. However, using Crank-Nicolson in straight tube geometry with various stabilization methods provides better accuracy than other second-order time stepping schemes, such as BDF-2. The type of stabilization method used when d < 10 nm does matter since Streamline Upwind Petrov-Galerkin introduces higher deposition fraction compared to Galerkin Least Square. This statement is not true when d > 10 nm, since mesh refinement is important at this range. In the human airways simulation, we found that for d = 1 nm the concentration distribution is uniform compared to d = 150 nm , where localized concentration exists. This implies a potential health risk when inhaling nanoparticles because nanoparticles have a very high surface area and the potential for exposure is much greater. The stabilization methods tested in this thesis show promise for modeling nanoparticle transport in the human airways.