Implementation of the Raschi model

In order to better evaluate the quality of the 3D SFDM, it is compared against the model by Raschiet al.[55]. This model is summarized by the following equations:



where Re_his the Reynolds number based on the hydraulic diameterd_h, which is defined as four times the ratio of the connected void volume to the wetted surface areaA_w.

d_h=4V_void

A_w (4.9)

Raschi model is isotropic in all directions. For the tangential stressesτ_yandτ_z, the velocity u_xjust needs to be replaced byu_yandu_z.

For this model, we compute the local porosity in the same way as for the SFDM, whereas in the Raschi’s article [55], a uniform porosity for the whole stent is used. The model also depends on the hydraulic diameterd_h, which is different from the strut diameter used in the SFDM. Figure4.14illustrates the procedure we adopted to compute an approximate value of the hydraulic diameter, by computing a projection of the stent.

Fig. 4.14 The illustration for the computation of hydraulic diameter for Raschi’s model In this sketch,dis the diameter of strut,A₀andA₁are the areas enclosed between the red lines and blue lines, which represent the center lines of the struts. According to Equation4.9, V_void and A_w are required for the computation ofd_H. V_void is approximated by the product between the void projection areaA₀ andd. To computeA_w, we consider the 3D solid parts enclosed in the blue rhombic areas as half of a straight cylinder with diameterdand lengthL.

Then from the information above, we conclude:















A₀/A₁=β V_void =A₀d A₁−A₀= ^d₂L A_w= ^d₂Lπ

(4.10)

From this, the hydraulic diameter is computes as:

d_H= 4V_void

A_w = 4dβ

(1−β)π (4.11)

4.5 Results 75 Patient 2 Patient 3 Patient 4

Fine resolutiondx(µm) 28.718 18.338 20.975

Time step(F)dt(s) 6.2157×10⁻⁶ 2.9522×10⁻⁶ 1.9628×10⁻⁶

Coarse resolutiondx(µm) 86.153 55.015 73.414

Time step(C)dt(s) 1.8647×10⁻⁵ 8.8565×10⁻⁶ 6.8698×10⁻⁶

Resolution ratio(C/F) 3.0 3.0 3.5

Table 4.2 The resolution for all simulations

4.5 Results

In this section, the results of simulations of a fully resolved stent are compared with coarse-grained simulations using our SFDM and the porous-media model by Raschiet al. The fully resolved simulation used the fine resolution, which was determined through a grid convergence study in section4.3. The other two models were executed at a coarser resolution, with a scale factor of 3 in every space direction for Patient 2 and Patient 3, and a scale factor of 3.5 for Patient 4. These scale factors were experimentally found to be optimal, because at higher scale factors, significant differences between the fully resolved and coarse-grained simulations occur.

All resolutions are summarized in Table4.2.

All comparisons between fully resolved and coarse-grained simulations are carried out at the point during the cardiac cycle when the flow rate is highest, as indicated by a red point for each patient case in Figure4.8. It should be mentioned that for technical reasons, the exact maximum was slightly missed in the case of Patient 4 (see Figure4.8), because our error analysis was performed in a post-processing step, using voluminous simulation data produced at specific time intervals only.

In the following, the results obtained in the patient cases 2, 3, and 4 are be presented and analyzed individually, but the figures are integrated together for the reason of space.

4.5.1 Patient 2

Figure4.15aprovides a back and a side view of the iso-velocity surface atV =0.05m/sfor Patient 2 test case. The images focus on the aneurysm, which for the investigated medical problems is the region of interest. Clearly, the SFDM reproduces the iso-velocity surface of the fully resolved simulation quite precisely, while in the Raschi model, the iso-surface extends too far in direction of the aneurysm wall.

Figure4.17adisplays, at the same moment in the cardiac cycle, iso-contours of the velocity norm on two slices through the aneurysm (the position of the slices is shown in Figure4.16a. In Slice 1, the iso-velocity lines of the SFDM are generally similar to those of the fully resolved

simulation. Slice 2 shows that the SFDM does not exhibit substantial differences in the region close to the right wall, while on the left side, the velocity contour of the vortex at 84mm/s is smaller than the one of the fully resolved simulation (with a maximum of 112mm/s). But the SFDM still presents velocity pattern very close to the one of the fully resolved stent. The Raschi model on the other hand exhibits much denser iso-velocity lines near the wall, and the velocities inside the aneurysm are substantially too high.

(a) Patient 2 (b) Patient 3

(c) Patient 4

Fig. 4.15 Iso velocity surface with back view (top) and side view (bottom). From left to right, the results of the fully resolved stent, the SFDM, and the Raschi model are shown.

(a) Patient 2 (b) Patient 3 (c) Patient 4

Fig. 4.16 Position of the slices. Right: Slice 1. Left: Slice 2.

4.5 Results 77

(a) Patient 2

(b) Patient 3

(c) Patient 4

Fig. 4.17 Velocity norm on two slices inside the aneurysm

Figures4.18aand4.19acompare the simulations quantitatively over the first two cardiac cycles. Except for a short initial transient behavior, the data in both cardiac cycles is exactly identical, an observation that will be exploited to restrict the computationally more intensive case of Patient 4 to a single cardiac cycle. Figure4.18adisplays the WSS in the four measure-ment points (defined in Figure4.9). The SFDM shows some discrepancy in point A and B from the fully resolved stent simulation, while Point C and D correspond very well. The most important discrepancy is observed in point A. This point is however located on the aneurysm neck, very close to the stent, and it is not unexpected that in this area discrete properties of the fully resolved stent have a noticeable effect on the flow. The results of the Raschi model on the other hand deviate substantially from the fully resolved simulation.

Figure4.19ashows the error of the values predicted by the SFDM, for the average, the minimum, and the maximum value of the WSS during the second cardiac cycle. We don’t show the errors for the Raschi model, as they are substantial and would impact the readability of the figure too much. The error is defined as the difference between SFDM and fully resolved simulation, divided by the value for the fully resolved simulation. As expected, the largest errors are in Point A, which is situated on the aneurysm neck. In all other points, the errors are below 20%, and for the maximum WSS, a critical parameter, even below 5%.

4.5.2 Patient 3

Figure4.15bshows the iso-contour of the velocity norm at 0.15m/sfor the test case Patient 3. The fully resolved simulation predicts the formation of a dome, the shape of which is underestimated by the SFDM, while the Raschi model overreaches its extent. It can be concluded that the SFDM overestimates the drag forces generated by the stent, while the Raschi model underestimates them. Figure4.17bshows the iso-contours of the velocity on two slices through the aneurysm. On Slice 1, which has a larger cross-sectional area, the SFDM produces results similar to the fully resolved simulation, but on slices with a smaller cross-sectional area like Slice 2, the high-velocity contours (140mm/s) are not reproduced in bulk areas of the aneurysm and are only found in the region close to the neck. This further confirms that the SFDM slightly underestimates the velocity in some regions of the aneurysm, but the overall results remain relatively close to the predictions of the fully resolved simulation. The velocities generated in the aneurysm by the Raschi model, on the other hand, are distinctively too large, and even produce a qualitatively different pattern.

Figures4.18band4.19bpresent a quantitative comparison for Patient 3. It can be pointed out that in Point A, very close to the stent, the two force models, running in coarse simulations, produce large fluctuations in the WSS. The errors for the SFDM are evaluated over the second

4.5 Results 79

(a) Patient 2

(b) Patient 3

(c) Patient 4

Fig. 4.18 WSS in the measurement points. F in the legend means fine resolution, C coarse resolution

(a) Patient 2 (b) Patient 3 (c) Patient 4

Fig. 4.19 The WSS errors between the SFDM and the fully resolved simulations, at the measurement points. The four groups 1-4 from left to right correspond to Points A-D

cardiac cycle, as for Patient 2. The minimum values are again subject to the largest errors, while the maxima are quite accurate, below a 10% error margin.

4.5.3 Patient 4

Figure4.15cshows the iso surface of the velocity norm at 0.10m/s. While the SFDM under-estimates the shape of the iso surface only slightly, the Raschi model overreaches the shape substantially. The iso contours on slices for the SFDM in Figure4.17cshow a similar phe-nomenon as for Patient 3. For large cross-sections like Slice 1, the SFDM produces a very similar velocity pattern as the fully resolved simulation. But for Slice 2, the high velocity contours for 210mm/s and 175mm/s are not reproduced by the SFDM, and the 140mm/s contour is smaller, indicating an overall lower velocity than in the fully resolved simulation.

We conclude from these observations that the SFDM overestimates the drag exerted by the stent on the flow. With the Raschi model, the velocity is again substantially too high, as it can be clearly seen by the large circle formed by the 210mm/siso contour.

The WSS at Patient 4 measurement points for is shown in Figure4.18c. The match between SFDM and fully resolved simulation is very good, although some minor discrepancies are visible in Points B and D. Figure4.19cshows the error of the average, maximum, and minimum WSS, which, given that we only simulated one cardiac cycle for Patient 4, were computed in the time interval from 0.5sto 1.5sto exclude the initial transients. As before, the biggest differences show up in Point A, which is on the neck of the aneurysm. Generally speaking, the errors are all very acceptable, especially the errors of the maximum values, which are almost negligible.