• nanoparticle;
  • transport;
  • finite element;
  • stabilization


  1. Top of page

The transport and deposition properties of nanoparticles with a range of aerodynamic diameters ( 1nm ⩽ d ⩽150nm) were studied for the human airways. A finite element code was developed that solved both the Navier–Stokes and advection–diffusion equations monolithically. When modeling nanoparticle transport in the airways, the finite element method becomes unstable, and, in order resolve this issue, various stabilization methods were considered in terms of accuracy and computational cost. The stabilization methods considered here include the streamline upwind, streamline upwind Petrov–Galerkin, and Galerkin least squares approaches. In order to compare the various stabilization approaches, the approximate solution from each stabilization approach was compared to the analytical Graetz solution, which is a model for monodispersed, dilute particle transport in a straight cylinder. The optimal stabilization method, especially with regard to accuracy, was found to be the Galerkin least squares approach for the Graetz problem when the Péclet number was larger than 10 4. In the human airways geometry, the Galerkin least squares stabilization approach once more provided the most accurate approximate solution for particles with an aerodynamic diameter of 10 nm or larger, but mesh size had a much greater effect on accuracy than the choice of stabilization method. The choice of stabilization method had a greater impact than mesh size for particles with an aerodynamic diameter 10 nm or smaller, but the most accurate stabilization method was streamline upwind Petrov–Galerkin in these cases. Copyright © 2013 John Wiley & Sons, Ltd.


  1. Top of page

Nanoparticles, both harmful and therapeutic, can be inhaled into the human airways and be deposited throughout the respiratory tract. The transport and deposition location of nanoparticles are determined by particle characteristics, air flow rate, and airway geometry [1-5]. Because nanoparticles have a large surface area per unit mass, the potential intrinsic toxicity is increased, and, if the lungs clear the nanoparticles slowly, it also increases the exposure time. Therefore, nanoparticles that are not biocompatible could be very damaging to the lungs, as demonstrated by the example of the lung disease mesothelioma, caused by prolonged exposure to asbestos [1]. Because of the numerous experimental challenges associated with measuring the location of nanoparticle deposition in the human airways, the development of an efficient and accurate computational modeling approach is crucial.

There have been many theoretical studies performed on microparticle transport (i.e., particles with an aerodynamic diameter larger than 1  μm) in the human airways, but very few studies have focused on nanoparticles (i.e., aerodynamic diameter less than 500 nm) in human airways [1-4, 6-10]. Particles with aerodynamic diameters larger than 1 micron are typically modeled individually using a particle-tracking algorithm, but this is not practical for nanoparticles because of the large number of individual particles and the greater impact of Brownian motion on the particle trajectories. Nanoparticles are frequently modeled using a continuum approach in which a small range of particle sizes is approximated as monodispersed, and a single diffusivity value is used for a particular size range. The continuum approach is based on the advection–diffusion equation and is computationally challenging because of numerical stability issues, especially when using the Galerkin FEM. Modeling the human airway poses several other challenges because the exact geometry is not known because of the large number of branching generations [1, 3, 11]. Previous studies have presented various airway geometries with varying levels of geometric complexity [5, 6, 12-15]. The Weibel model was the very first widespread model that gave geometric information such as airway diameter and length for the lung, and the Weibel Type A model remains a popular choice for describing symmetrically bifurcating airways [1, 3, 5, 6, 17-19] even though other studies have shown that the Weibel model is often not accurate [20]. In this paper, we use a very simple model geometry of the human airway (G0–G3), which is based on Weibel's model [21]. Another method for obtaining human airway geometry data is to use a computed tomography scan or MRI, but these experiments are very difficult and expensive to conduct [4, 7, 18]. A particularly interesting problem, which is not addressed in this paper, is the change in human airways geometry and transport properties due to disease.

As is demonstrated later in Section 3, the Galerkin FEM is not stable for practical mesh sizes when modeling nanoparticle transport because of the small magnitude of the diffusivity coefficient relative to the velocity (i.e., the high Péclet number). The standard Galerkin FEM approach gives an approximate solution with nonphysical oscillations at the nodal level [22, 23]. Accurately solving the model equation systems with a Galerkin FEM discretization normally requires one of several different stabilization methods (e.g., the streamline upwind Petrov–Galerkin [SUPG] method) [22, 24-28]. A few alternatives to the stabilized Galerkin FEM approach used here have been used to model advection dominated nanoparticle transport. For example, the DG method has been applied to advection dominated problems, but, while this method is stable, it is difficult to implement in 3D and computationally expensive [29]. Another alternative method is the finite volume method (FVM), but it can be difficult to obtain a higher-order approximation making the method computationally expensive for some problems when a high accuracy approximation is desired [22, 23]. The higher-order approximation that is possible with the FEM framework employed in this study can capture Brownian motion better than the alternative lower-order methods [23, 30]. Our goal in this work is to evaluate and compare various stabilization methods for the Galerkin FEM applied to modeling nanoparticle transport in both an idealized geometry using the Graetz solution [31] and for human airway geometry with multiple bifurcations. The focus is on comparing the accuracy and computational costs of the various stabilization methods and establishing the best choice for modeling nanoparticle transport.


  1. Top of page

Navier–Stokes and advection–diffusion

Several phenomena in fluid mechanics can be described using the Navier–Stokes equations [22, 23, 32, 33]. Using the Navier–Stokes equations, we can theoretically model both turbulent and laminar fluid flow, but in this paper, we will focus on a constant inhalation flow rate of 15 L/min, which will only generate laminar flow [2, 5, 7, 34]. Many studies have shown that the incompressible Navier–Stokes equations are appropriate when modeling laminar flows in the bifurcating human airways, even though air is compressible, due to the low Mach number [1, 3, 4, 17].

For an incompressible fluid, the continuity equation is

  • display math(1)

where inline image is the velocity. The conservation of momentum equation for a Newtonian fluid is given by

  • display math(2)

where ρ is the density, p is the pressure, η is the viscosity, and σ is the strain tensor, which is defined as inline image. Nanoparticle transport for a single particle diameter is described by the advection–diffusion equation.

  • display math(3)

In Equation (3), Cis the monodispersed particle concentration, and D is the diffusion coefficient. It has been well established that the Galerkin FEM can be unstable for advection dominated problem such as nanoparticle transport in the human airways due to spurious oscillations [3, 22, 23]. In order to solve this problem, we must add a stabilization term (often referred to as upwinding) into the standard Galerkin weak form. After Equation (3) is nondimensionalized, the weak form of the equation is derived by multiplying the advection–diffusion equation by a test function, integrating each term, and integrating any second-order derivative terms by parts [22, 23, 32, 33]. Using w for the test function, the weak form for the advection–diffusion equation is

  • display math(4)

The inverse Péclet number is defined as 1 ∕ Pe = D ∕ v0L and can be calculated using the diffusion coefficient (D), characteristic length (L), and characteristic velocity (v0). The diffusion coefficient for the nanoparticles in the current airway model was based on a previous study [7] and is summarized in Table 1. The boundary conditions used for the Navier–Stokes and advection–diffusion equations for both the Graetz problem and the human airways simulation are described in Section 3.

Table 1. Relationship of particle diameter and diffusion.
Particle diameter (nm)Diffusion D(m2 ∕ s)


Different stabilization methods are evaluated here for their accuracy and computational efficiency on both the Graetz problem and the inhaled nanoparticles in the human airway problem. While a number of stabilization techniques have been developed, the technique with the simplest form that is considered here is the SU stabilization method, and the other stabilization methods will be presented as extensions to it. The SU stabilization method adds diffusion in the flow direction and not transversely using a diffusion operator in tensorial form [22]. The final form of the SU method can be viewed as the standard Galerkin weak form for the advection–diffusion equation (with the boundary term omitted) plus an additional stabilizing term:

  • display math(5)

In Equation (5), τ is the intrinsic time, which is a constant typically between 0 and 1 but can be larger than 1. Many researchers have developed methods for estimating the τ parameter [22, 24-26, 35-37]. Instead of using one of these methods (see [38, 39]), the impact of τ on the accuracy of the approximation method is evaluated for the problems of interest in Section 3. The SU method utilizes the upwind test function only for the advection term, so there exists a potential problem regarding the accuracy of this method because the method adds diffusion along the streamline in an inconsistent manner [22, 23, 36, 37, 40]. A more consistent stabilization method is needed, and, extending from the SU method, the SUPG method was created.

The SUPG method is very similar to SU; the only difference is an additional term in the weak form that makes the method consistent [24, 25]:

  • display math(6)

Many studies have shown that the SUPG method performs better than SU [22]. Some forms of the SUPG method for the advection–diffusion equation include a higher-order term, ( ∇ · ((1 ∕ Pe) ∇ C)), but this term vanishes when using linear elements [22, 28]. This simplification is also used here but is only an approximation for the quadratic elements employed in the following equations.

The Galerkin least squares (GLS) method addresses a potential stability issue with the SUPG method under advection dominated conditions because it adds a symmetric stabilization term consistently [22]. The GLS method works by element-to-element, weighted least squares of the advection–diffusion equation, and this method provides symmetry, which is important for achieving stability [22]. However, caution must be used when using the GLS method when 1 ∕ Pe is not small (i.e., advection term is not dominating and stabilization is probably not necessary) because GLS will tend to amplify some error modes more than SUPG as the GLS method adds a Galerkin weighting term to SUPG [22, 41, 42]. In Equation (7), we can see the additional terms added to the advection–diffusion equation:

  • display math(7)

Here, dt is the time step size. The term w ∕ dt in Equation (7) can be seen as an approximation to the term one would see in a GLS formulation in the context of a space–time formulation [27, 41, 42]. Such a term in the context of a space–time formulation can also be seen as part of an SUPG stabilization term (see [27]), because the time derivative can be seen as part of an augmented advective operator.

Deposition fraction parameters

In order to quantify the amount of particulate that diffuses to the wall, boundary conditions must be implemented for the geometry of interest. The airway walls act approximately as a sink, so the boundary condition at the wall is a Dirichlet type boundary condition where the concentration is set to zero. This boundary condition is a good assumption for the nearly instantaneous reaction kinetics between the wall and common particulate [3, 4, 7]. At the inlet, a paraboloid with a peak of one is set for the concentration, and, at the outlet, a steady flow, Neumann boundary condition is applied. Because the airway wall concentration is set to zero, it is straightforward to calculate the amount of deposited particulate or the deposition fraction (DF) in an airway region by determining the total mass flux difference between the inlet and the outlet [2, 4, 5]. The DF is calculated using

  • display math(8)

where Ai is the area of the boundary face for element i, n is the total number of element faces on the boundary, Ci is the average concentration for the element face, and Vi is the average normal velocity for the face.

Numerical methods

The simulation of particle transport in the human airways requires approximating the solution to both the Navier–Stokes and advection–diffusion equations, and we opted to solve this system of equations monolithically (i.e., simultaneously). The spatial discretization method used is the FEM, and all algorithms are implemented in C++. A number of external libraries were also used to support the algorithms that were developed in-house. Matrix and vector data structures are those from Epetra, which is part of the Trilinos package from Sandia National Lab [43, 44]. Linear solvers are also from the Trilinos package, which provides a number of different solvers including direct and iterative solvers. The elements used for the spatial discretization are 10-node/4-node tetrahedron, which is a Taylor–Hood element (P2P1) and satisfies the inf-sup condition for the Navier–Stokes equations [23, 45, 46]. Because our focus for this model is at very low Reynolds number (Re), stabilization for Navier–Stokes is unnecessary, and application to high Re will be our focus for future work. The finite element mesh was spatially decomposed using METIS so that the problem could be executed in parallel on a distributed memory platform using the message passing interface (MPI) [47].

We implemented two different second-order accurate methods for temporal discretization: the Crank–Nicolson (CN) and the second-order backward difference formula (BDF-2) discretization [48]. The stabilization methods (SU, SUPG, and GLS) were validated using the Graetz solution, which is a steady-state solution to the advection–diffusion equation in a straight cylinder. The human airways (G0–G3) simulation was run for varying particle diameter (1 to 150 nm), and the DF values were calculated to determine the amount of particulate diffusing to the wall of the airways.

The finite element mesh for all test problems was created using the software package CUBIT, which is developed and maintained by Sandia National Laboratory. An example, meshed geometry is shown in Figure 1, and we used an unstructured mesh to better approximate the geometry using surface conforming tetrahedron. The meshing algorithm created more elements near the walls, inlet surface, and outlet surface because the larger gradients are found in those locations. We refined the mesh till the DF value converged. The finest mesh had approximately 110,000 elements for a G0–G3 human airway. The simulations were performed on a Dell workstation with dual 6-core Xeon processors. The convergence criteria for both GMRES iterations and Newton iterations were set at a tolerance of 10 − 8 [23, 45]. The computational time for the human airways simulation varied with the number of elements in the mesh, and the finest mesh, which had approximately 100,000 elements, took 72–120 h of computational time to simulate 500 time steps, with a step size of 0.1 (dimensionless time).


Figure 1. Mesh refinement at G0–G3 bifurcation with 10-node tetrahedron.

Download figure to PowerPoint

Model validation

The stabilization methods used for the human airways model are first compared using the analytical Graetz solution from previous study [31]. The validation was conducted on a cylindrical geometry, and the boundary conditions used were based on the analytical Graetz solution. The model has been tested extensively by varying the diffusion coefficient and maintaining a constant Re of 1.0 for all validation tests using the Graetz solution. The accuracy of the model was determined by computing the L2-norm of the error.


  1. Top of page

Graetz solution

The Graetz problem is an important benchmark problem in the study of advection–diffusion problems because it has an analytical solution, and it shares a number of important characteristics with the human airway problem that is of primary interest here. The Graetz solution can be used to compare the accuracy of different stabilization methods and determine the approximation error associated with each method in any desired norm. The Graetz problem can also be used to determine the optimal τ value for a particular stabilization method. Determining a single, optimal τ value can be simpler than implementing an element based method for determining τ, which has been proposed by other researchers [28, 40, 49].

The analytical solution to the Graetz problem was obtained from previous work [31]. A cylindrical mesh, as shown in Figure 2, was constructed with approximately 4500 elements. Varying the number of elements to a coarser or finer grid will impact the optimal τ value, but this change is avoided by fixing the number of elements in the mesh. The boundary conditions for the Graetz problem are summarized in Table 2, and finally, the Re was set to 1.0. Because previous studies have shown that the time step size Δt does have an impact on the optimal τ coefficient, this relationship is not studied in detail, and, for these simulations, Δt was set to 0.01 [22, 36, 37, 40]. Note that both time discretization methods tested here are implicit methods, so any time step size is temporally stable.


Figure 2. Meshed cylinder with 4532 elements, which was used for the Graetz solution comparison.

Download figure to PowerPoint

Table 2. Boundary condition for the Graetz problem.
InletGraetz solution1.0 − (x2 + y2)
Outletn · (1 ∕ Pe) ∇ C = 0.0 − p + μ (n · ∇ )u = 0.0

The inlet boundary condition for concentration was set as the Graetz solution. Because we are solving the model equations monolithically (i.e., in a fully coupled manner), the Navier–Stokes solution in a cylinder will be close to the analytical solution (a paraboloid), and an error analysis of the accuracy of the Navier–Stokes solution is not necessary. The cylinder has a dimensionless radius (r), r = 1, and a dimensionless length (h), h = 10.

The comparison tests were conducted with varying values for the inverse Péclet number, 1 ∕ Pe = 0.01 to 0.000001 (1 to 50 nm particles) and also varying values for τ (0 to 1.0). The standard Graetz problem has a discontinuity in the concentration field along the edge of the cylinder where the inlet surface and wall meet. This is due to the common approach of setting the inlet concentration to 1.0 and the wall concentration to 0.0. As a result of these two boundary conditions, there is a discontinuity along the edge of the inlet, and this discontinuity is incompatible with the continuous finite element space used for the approximate solution. To avoid the discontinuity, the cylindrical domain that was meshed was downstream of the original inlet, and the analytical solution to the Graetz problem was used as the inlet boundary condition for this downstream domain.

In Figure 3(a), we can see the growth of the error when the regular FEM method is used without any stabilization, and the inverse Péclet number is varied. This shows that stabilization methods are necessary when approximating the solution to the advection–diffusion equation with small ( < 10 − 3) values for 1 ∕ Pe. An example of an unstable result is shown in Figure 3(c), and, as will be shown, introducing stabilization will help reduce the instability error. When inline image, stabilization is not required because the regular Galerkin FEM provides a very good approximation, as shown in Figure 3(b). Another method for avoiding instability is to refine the mesh, but this introduces other difficulties such as extremely long computational times and increased capabilities for the meshing software [3]. Other studies have shown that as the number of elements increases in a mesh, τ[RIGHTWARDS ARROW] 0, and this is also what we have observed in this study [22]. This is because as the number of elements increases, the better the approximation we can obtain in the FEM space as long as no discontinuities exist in the solution or its derivatives.


Figure 3. (a) L2-error for FEM without any stabilization at various inverse Péclet numbers, (b) cylinder slice at Y = 0 for 1 ∕ Pe = 10 − 2, and (c) 1 ∕ Pe = 10 − 6, which shows the numerical instability.

Download figure to PowerPoint

In Figure 4, we can see the relationship between the L2-error and τ for the different stabilization and temporal discretization methods examined here. Figure 4 displays a comparison of the different stabilization methods for different values of τ and for a fixed 1 ∕ Pe = 10 − 5, all the stabilization methods were temporally discretized using the BDF-2 method except for the data points CN-SUPG, which used the CN method. The most accurate method for stabilization of the advection–diffusion equation for this particular problem and diffusivity value is the GLS method. When τ ≈ 0.3, the various stabilization methods show the smallest error as shown in Figure 3, and this is the optimal τ value found for the Graetz problem. The second best stabilization method is SUPG, and, finally, SU is the least accurate method. Even though there are many studies that have proposed various methods for calculating the stabilization parameter (τ), using a constant τ value reduces implementation time. In Figure 4, we can clearly see the impact of the different second-order temporal discretization methods. The stabilization methods do have an impact on the temporal accuracy, and using the CN method provides better accuracy than the BDF-2 method. We are also aware that previous studies have compared various stabilization methods with an exact solution in 1D and 2D, but to our best knowledge, this is the first study that has compared various stabilization methods with the Graetz solution in 3D [22, 36, 37].


Figure 4. L2-error for Graetz problem (1 ∕ Pe = 10 − 5) using different stabilization methods at various τ values and different temporal discretizations. SU, streamline upwind; SUPG, SU Petrov–Galerkin; GLS, Galerkin least squares; CN, Crank–Nicolson.

Download figure to PowerPoint

Figure 5(a) displays a comparison of the error resulting from different stabilization methods for different values of 1 ∕ Pe when τ = 0.3. We can clearly see that the best method for stabilization when varying 1 ∕ Pe for this particular problem and τ = 0.3 is the GLS method. Cross-sectional slices along the axis of the cylinder are shown in Figure 5(b), (c), and (d) for the various stabilization methods, and the results show that all methods stabilize the solution and give qualitatively similar solutions. The GLS stabilization method captures the physics better than the other stabilization methods when 1 ∕ Pe ⩽10 − 4, as shown in Figure 5(a). When inline image, the GLS method amplifies the instabilities from regular FEM, which has been shown in previous studies [22]. When 1 ∕ Pe ⩽10 − 3, both SU and SUPG methods provide similar performance.


Figure 5. L2-error at (a) various inverse Péclet numbers and cylinder slices at Y = 0 for 1 ∕ Pe = 10 − 6 for (b) streamline upwind (SU)-FEM, (c) SU Petrov–Galerkin (SUPG)-FEM, and (d) Galerkin least squares (GLS)-FEM.

Download figure to PowerPoint

We also implemented an element-level calculation for τ, which has been proposed in [26, 50]. The results are shown in Figure 6 where the element-level τ (TF-SUPG) is compared with a constant τ = 0.3 for the Graetz problem. We see that the element-level calculation of τ does provide better accuracy compared to scalar τ, but this additional calculation increases implementation complexity and computational time by a small (less than 10%) amount.


Figure 6. L2-error as function of 1 ∕ Pe for streamline upwind Petrov–Galerkin (SUPG) stabilization method with an element-level calculation for τ(TF–SUPG) and constant value of τ = 0.3.

Download figure to PowerPoint

Human airways simulation

For the human airway simulation of nanoparticle deposition, there is no analytical solution, and the comparison of the various stabilization methods is based on the DF instead. The impact of only two different stabilization methods, SUPG and GLS, is compared because the third stabilization method, SU, provides very similar performance to SUPG. Many of the choices made in the human airway model are motivated by the observations made from the previous Graetz problem.

The simulations for human airways were run with a constant inlet volumetric flow rate of Q0 = 15 L/min. This flow rate impacts the (Re), so we need to calculate the Re based on the characteristic velocity, that is, the peak inlet velocity (v0), and the characteristic length, that is, the diameter of the first generation (L). After calculating these variables, Re = 200 is obtained and used in all the simulations. In this work, the volumetric flow rate and Re are not varied to larger values because a higher Re would require stabilizing the Navier–Stokes equations or using a significantly finer mesh. Stabilization of the Navier–Stokes equations creates additional complexity because there would be two equations that require stabilization. The geometry was meshed with 10-node tetrahedron, and this step was performed using the CUBIT software. The mesh has only one inlet and six outlets.

The boundary conditions are listed in Table 3 for both velocity and concentration for the inlet, the outlet, and the wall. These boundary conditions are consistent with those used in other studies [2, 3, 7]. The simulation was run for particle diameters ranging from 1 to 150 nm (the corresponding diffusivities values are given in Table 1), and the simulations used SUPG and GLS stabilizations. The stabilization methods were compared by calculating the flux at both the inlet and the outlet of the airways. The flux is then used to calculate the DF, defined by Equation (7).

Table 3. Boundary conditions for human airways simulation.
Inlet(1 − y) (1 + y) (1 − x) (1 + x)2 * ((1 − y) (1 + y) (1 − x) (1 + x))
Outletn · (1 ∕ Pe) ∇ C = 0.0 − pn + μ (n · ∇ )u = 0.0

Because modeling human airways is computationally expensive, as stated previously, the model was developed to execute in parallel in a distributed memory environment. For any computational model, the scalability becomes an important aspect of the algorithm development, and it can also show how efficiently the model has been implemented. In order to show the scalability, multiple human airway meshes with varying numbers of elements were created. The simulation was run to t = 500 (dimensionless time) to ensure that every solution had reached steady state, and all the simulations were run with four processors. Figure 7 shows the scalability of the algorithm, and we can see that for a range of problem sizes, there exist an approximately linear relationship between computational time and number of elements using preconditioned iterative method (AztecOO–Generalized minimal residual (GMRES) method).


Figure 7. Computational time and deposition fraction (DF) relationship with number of elements in a mesh.

Download figure to PowerPoint

In order to examine the accuracy of the various stabilization approaches, a similar approach to the scalability test was employed, and the DF values were calculated for varying numbers of elements. A base case was used for all the simulations run where SUPG was used as the stabilization method, and the diffusion was set to 1 ∕ Pe = 10 − 5. Once again, the simulation was run until t = 500 (dimensionless) to ensure that the solution had reached steady state. The DF value was monitored until it converged to a single value, and this is used as the criteria for steady state. Figure 7 shows the DF results for meshes with varying number of elements, and we can conclude that as the mesh becomes finer, the DF value converges to a nonzero value, as expected. This trend reflects the improvement in accuracy when more elements are used in the simulation.

The next set of simulations was run for particles with different diameters, and these runs were conducted with three different mesh resolutions. The mesh resolutions that were used had approximately 30,000, 80,000, and 110,000 elements for comparison between coarse and fine meshes. In Figure 8, we can see changes in the DF with respect to particle diameter for both SUPG and GLS stabilization (note that ‘*’ denotes simulations run on the finest mesh and ‘ ∼ ’ denotes the mesh with 80,000 elements).


Figure 8. Deposition fraction (DF) as function of particle diameter for a 30,000 element mesh for streamline upwind Petrov–Galerkin (SUPG) and Galerkin least squares (GLS) stabilizations; SUPG* and GLS* are simulations with 100,000 elements.

Download figure to PowerPoint

There are several important relationships to take from Figure 8. First, we can see that for larger particle diameters (d > 10nm), the stabilization method that is chosen has minimal impact on the DF, and this holds true even when the mesh is refined. Consistent with the results from the Graetz problem, the GLS stabilization method has a small accuracy advantage for the larger particles (i.e., smaller diffusivity values) compared to SUPG. Second, for d < 10nm, the stabilization method that is used for the model becomes very important. In this case, the GLS method is not as accurate as the SUPG method, which is consistent with observations from the Graetz problem. The stabilization method used to stabilize the advection–diffusion equation does impact the spatial accuracy of the approximate solution. The impact of the choice of stabilization method as a function of particle diameter for human airways simulation has not been considered in other studies, because other studies primarily employ the FVM that is commonly available in commercial software [4, 5, 7, 51]. The FVM method is difficult to extend to higher-order approximations, so the only way to improve accuracy is to increase the number of elements in a mesh, which increases the computational time [22, 23]. The FEM is straightforward to implement with a higher-order approximation but requires stabilization for most particle sizes. The focus here on comparing stabilized FEM sets apart the current study form other studies on modeling nanoparticles in human airways.

Figure 9 shows the comparison of DF values for varying particle diameters using both an element-level calculation of τ based on [26, 50] and using a constant value (τ = 0.3). The comparison was conducted on a mesh with approximately 80,000 elements, and the results show that using the constant value for the stabilization parameter obtained from the Graetz solution (τ = 0.3) is indeed a good choice for modeling nanoparticle in human airways (1 to 150 nm).


Figure 9. Comparison of deposition fraction (DF) values for element-level τ(SUPG) and scalar τ = 0.3(SUPG ∼ ), with streamline upwind Petrov–Galerkin (SUPG) stabilization method.

Download figure to PowerPoint

Figure 10 shows the concentration distributions for nanoparticles using the SUPG stabilization method. The more diffusive 1 nm particle (Figure 10(a)) becomes deposited much more rapidly and uniformly compared to the 150 nm particle (Figure 10(b)). We can conclude that the nanoparticle DF will decrease when we increase the particle diameter (up to a point), and this is due to reduced diffusivity. In Figure 10(b), we can see that for particles with larger diameter, the dimensionless concentration distribution is less uniform, and there are significant differences in the concentration in the various outflow branches. The smaller particles in Figure 10(a) are more uniformly distributed among the various outflow branches. Re also has an impact on the DF values by varying the inlet flow rate, but this work does not focus on this impact.


Figure 10. Concentration distribution for (a) d = 1 nm and (b) 150 nm with streamline upwind Petrov–Galerkin stabilization method.

Download figure to PowerPoint


  1. Top of page

This paper evaluates a number of different stabilization methods that could be used to model nanoparticle transport in human airways. Specifically, the SU, SUPG, and GLS stabilization methods were compared using the Graetz problem and nanoparticle transport in the human airways. The Graetz problem was chosen because it has an analytical solution, so the exact error in the numerical method can be determined. Using the Graetz problem, it was found that stabilization is necessary for problems where 1 ∕ Pe < 10 − 3, and the smallest error was found using GLS stabilization. For the human airways problem, it was shown again to be necessary to use stabilization for 1 ∕ Pe < 10 − 3, and the GLS stabilization approach once again displayed the highest accuracy for advection dominated problems. For both the Graetz and human airway problems, GLS stabilization should not be used for 1 ∕ Pe > 10 − 3, and either SUPG or no stabilization should be used instead. It is important to note that the choice of stabilization method or no stabilization at all is much more important for smaller, more diffusive particles than for larger particles.


  1. Top of page

This work was supported by the NSF grant CBET-1249950 and the Flight Attendant Medical Research Institute. We would like to thank the anonymous reviewer for helpful suggestions and insights.


  1. Top of page
  • 1
    Hogberg SM. Modeling nanofiber transport and deposition in human airways. Department of Apllied Physics and Mechanical Engineering, Lulea University of Technology, 2010.
  • 2
    Kleinstreuer C, Zhang Z. Airflow and particle transport in the human respiratory system. Annual Review of Fluid Mechanics, 2010; 42: 301334.
  • 3
    Wininger CW, Heys JJ. Particle transport modeling in pulmonary airways with high-order elements. Mathematical Biosciences 2011; 232(1):1119.
  • 4
    Zhang Z, Kleinstreuer C, Kim CS. Airflow and nanoparticle deposition in a 16-generation tracheobronchial airway model. Annals of Biomedical Engineering 2008; 36(12):20952110.
  • 5
    Zhang Z, Kleinstreuer C, Kim CS. Comparison of analytical and CFD models with regard to micron particle deposition in a human 16-generation tracheobronchial airway model. Journal of Aerosol Science 2009; 40(1):1628.
  • 6
    Li Z, Kleinstreuer C, Zhang Z. Simulation of airflow fields and microparticle deposition in realistic human lung airway models. Part II: particle transport and deposition. European Journal of Mechanics - B/Fluids 2007; 26(5):650668.
  • 7
    Shi H, Kleinstreuer C, Zhang Z, Kim CS. Nanoparticle transport and deposition in bifurcating tubes with different inlet conditions. Physics of Fluids 2004; 16(7):21992213.
  • 8
    Zhang Z, Kleinstreuer C. Airflow structures and nano-particle deposition in a human upper airway model. Journal of Computational Physics 2004; 198(1):178210.
  • 9
    Zhang Z, Kleinstreuer C, Donohue JF, Kim CS. Comparison of micro- and nano-size particle depositions in a human upper airway model. Journal of Aerosol Science 2005; 36(2):211233.
  • 10
    Zhang Z, Kleinstreuer C, Kim CS. Micro-particle transport and deposition in a human oral airway model. Journal of Aerosol Science 2002; 33(12):16351652.
  • 11
    Kleinstreuer C. ‘Effects of asymmetric branch flow rates on aerosol deposition in bifurcating airways’, by Z. Zhang, C. Kleinstreuer and C. S. Kim - Response. Journal of Medical Engineering & Technology 2001; 25(3):127127.
  • 12
    Comer JK, Kleinstreuer C, Kim CS. Flow structures and particle deposition patterns in double-bifurcation airway models. Part 2. Aerosol transport and deposition. Journal of Fluid Mechanics 2001; 435: 5580.
  • 13
    Comer JK, Kleinstreuer C, Zhang Z. Flow structures and particle deposition patterns in double-bifurcation airway models. Part 1. Air flow fields. Journal of Fluid Mechanics 2001; 435: 2554.
  • 14
    Li Z, Kleinstreuer C, Zhang Z. Simulation of airflow fields and microparticle deposition in realistic human lung airway models. Part I: airflow patterns. European Journal of Mechanics - B/Fluids 2007; 26(5):632649.
  • 15
    Liu Y, So RMC, Zhang CH. Modeling the bifurcating flow in a human lung airway. Journal of Biomechanics 2002; 35(4):465473.
  • 16
    Li Z, Kleinstreuer C, Zhang Z. Particle deposition in the human tracheobronchial airways due to transient inspiratory flow patterns. Journal of Aerosol Science 2007; 38(6):625644.
  • 17
    Choi L-T. Simulation of fluid dynamics and particle transport in realistic human airways. School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, 2007.
  • 18
    Kleinstreuer C, Zhang Z, Li Z, Roberts WL, Rojas C. A new methodology for targeting drug-aerosols in the human respiratory system. International Journal of Heat and Mass Transfer 2008; 51(23-24):55785589.
  • 19
    Tronde A. Pulmonary drug absorption. In Pharmacy, Uppsala University, Uppsala, 2002; 186.
  • 20
    Sauret V, Halson RM, Brown IW, Fleming JS, Bailey AG. Study of the three-dimensional geometry of the central conducting airways in man using computed tomographic (CT) images. Journal of Anatomy 2002; 200(2):123134.
  • 21
    Weibel ER. Morphometry of the Human Lung. Academic Press: New York, 1963.
  • 22
    Donea J, Huerta A. Finite Element Method for Flow Problems. John Wiley & Sons Ltd.: Chichester, England, 2003.
  • 23
    Gresho PM, Sani RL, Engelman MS. Incompressible Flow and the Finite Element Method, Vol. 1. John Wiley & Sons Ltd.: Chichester, England, 2000.
  • 24
    Brooks AN, Hughes TJR. Streamline upwind Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering 1982; 32(1-3):199259.
  • 25
    Tezduyar TE, Park YJ. Discontinuity-capturing finite-element formulations for nonlinear convection-diffusion-reaction equations. Computer Methods in Applied Mechanics and Engineering 1986; 59(3):307325.
  • 26
    Tezduyar TE, Osawa Y. Finite element stabilization parameters computed from element matrices and vectors. Computer Methods in Applied Mechanics and Engineering 2000; 190(3-4):411430.
  • 27
    Tezduyar TE. Computation of moving boundaries and interfaces and stabilization parameters. International Journal for Numerical Methods in Fluids 2003; 43(5):555575.
  • 28
    Franca LP, Hauke G, Masud A. Revisiting stabilized finite element methods for the advective-diffusive equation. Computer Methods in Applied Mechanics and Engineering 2006; 195(13-16):15601572.
  • 29
    Rajaraman PK, Vo G, Heys JJ, Hansen G. Comparison of continuous and discontinuous Galerkin finite element methods for parabolic differential equations employing implicit time integration. Submitted.
  • 30
    Kleinstreuer C. Two-Phase Flow : Theory and Applications. Taylor & Francis: New York; London, 2003. xiv.
  • 31
    Tyree C, Allen J. Diffusional particle loss upstream of isokinetic sampling inlets. Aerosol Science and Technology 2004; 38(10):10191026.
  • 32
    Reddy JN. An Introduction to the Finite Element Method, 2nd ed., McGraw-Hill series in mechanical engineering. McGraw-Hill: New York, 1993. xix.
  • 33
    Reddy JN, Gartling DK. The Finite Element Method in Heat Transfer and Fluid Dynamics. Boca Raton: CRC Press, 1994.
  • 34
    Kleinstreuer C, Zhang Z, Li Z. Modeling airflow and particle transport/deposition in pulmonary airways. Respratory Physiology & Neurobiology 2008; 163: 128138.
  • 35
    Hughes TJR, Tezduyar TE. Finite-element methods for 1st-order hyperbolic systems with particular emphasis on the compressible Euler equations. Computer Methods in Applied Mechanics and Engineering 1984; 45(1-3):217284.
  • 36
    Bochev PB, Gunzburger MD, Shadid JN. Stability of the SUPG finite element method for transient advection-diffusion problems. Computer Methods in Applied Mechanics and Engineering 2004; 193(23-26):23012323.
  • 37
    Burman E. Consistent SUPG-method for transient transport problems: stability and convergence. Computer Methods in Applied Mechanics and Engineering 2010; 199(17-20):11141123.
  • 38
    Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR. YZ beta discontinuity capturing for advection-dominated processes with application to arterial drug delivery. International Journal for Numerical Methods in Fluids 2007; 54(6-8):593608.
  • 39
    Hsu MC, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR. Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Computer Methods in Applied Mechanics and Engineering 2010; 199(13-16):828840.
  • 40
    Codina R. Stability analysis of the forward Euler scheme for the convection-diffusion equation using the SUPG formulation in space. International Journal for Numerical Methods in Engineering 1993; 36(9):14451464.
  • 41
    Tezduyar TE, Behr M, Liou J. A new strategy for finite-element computations involving moving boundaries and interfaces—the deforming-spatial-domain space-time procedure. 1. The concept and the preliminary numerical tests. Computer Methods in Applied Mechanics and Engineering 1992; 94(3):339351.
  • 42
    Tezduyar TE, Behr M, Mittal S, Liou J. A new strategy for finite-element computations involving moving boundaries and interfaces—the deforming-spatial-domain space-time procedure. 2. Computation of free-surface flows, 2-liquid flows, and flows with drifting cylinders. Computer Methods in Applied Mechanics and Engineering 1992; 94(3):353371.
  • 43
    Heroux MA, Bartlett RA, Howle VE, Hoekstra RJ, Hu JJ, Kolda TG, Lehoucq RB, Long KR, Pawlowski RP, Phipps ET, Salinger AG, Thornquist HK, Tuminaro RS, Willenbring JM, Williams A, Stanley KS. An overview of the Trilinos Project. Acm Transactions on Mathematical Software 2005; 31(3):397423.
  • 44
    Heroux MA, Sala M. The design of Trilinos. Applied Parallel Computing: State of the Art in Scientific Computing 2006; 3732: 620628.
  • 45
    Greenbaum A. Iterative Methods for Solving Linear Systems, Frontiers in Applied Mathematics 17. Society for Industrial and Applied Mathematics: Philadelphia, PA, 1997. xiii, 220 p.
  • 46
    Šolin P, Segeth K, Dolezel I. Higher-Order Finite Element Methods, Studies in advanced mathematics. Chapman & Hall/CRC: Boca Raton, FL, 2004. xx, 382 p.
  • 47
    Kumar GKaV. A software package for partitioning unstructured graphs, partitioning meshes, and computing fill-reducing orderings of sparse matrices, 1998.
  • 48
    Faires JD, Burden RL. Numerical Methods, 3rd ed. Thomson/Brooks/Cole: Pacific Grove, CA, 2003. xii, 622 p.
  • 49
    Tezduyar TE, Ramakrishnan S, Sathe S. Stabilized formulations for incompressible flows with thermal coupling. International Journal for Numerical Methods in Fluids 2008; 57(9):11891209.
  • 50
    Akin JE, Tezduyar TE. Calculation of the advective limit of the SUPG stabilization parameter for linear and higher-order elements. Computer Methods in Applied Mechanics and Engineering 2004; 193(21-22):19091922.
  • 51
    Kleinstreuer C, Zhang Z. Laminar-to-turbulent fluid-particle flows in a human airway model. International Journal of Multiphase Flow 2003; 29(2):271289.