#### 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 Δ*t* does have an impact on the optimal *τ* coefficient, this relationship is not studied in detail, and, for these simulations, Δ*t* 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.

Table 2. Boundary condition for the Graetz problem.Boundary | Advection–diffusion | Navier–Stokes |
---|

Inlet | Graetz solution | 1.0 − (*x*^{2} + *y*^{2}) |

Wall | 0.0 | 0.0 |

Outlet | *n* · (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 , 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* 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.

In Figure 4, we can see the relationship between the *L*_{2}-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 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 , 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.

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.

#### 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 *Q*_{0} = 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 (*v*_{0}), 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.Boundary | Advection–diffusion | Navier–Stokes |
---|

Inlet | (1 − *y*) (1 + *y*) (1 − *x*) (1 + *x*) | 2 * ((1 − *y*) (1 + *y*) (1 − *x*) (1 + *x*)) |

Wall | 0.0 | 0.0 |

Outlet | *n* · (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).

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).

There are several important relationships to take from Figure 8. First, we can see that for larger particle diameters (*d* > 10* *nm), 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* < 10* *nm, 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 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.