In this last case we consider the shape optimization of a femoro-popliteal bypass graft, as well as the robust optimization of this configuration with respect to residual flows through the occluded branch. Even if it is considered at the end of this chapter, the analysis of this problem – performed before the Navier-Stokes aorto-coronary case – has been the first attempt to the development of the current framework to robust optimization problems and to more general inverse problems.
In particular, we aim at solving the problems above by using a RB+FFD approach, comparing three cost functionals – the Stokes-tracking functional J2, the vorticity J3 and the one based on
Galilean invariant J4. Before setting the reduced framework, we performed a preliminary test,
in order to select pπ = 6 design parameters π1, . . . , π6 for constructing the FFD map T (·; π),
We have compared the strategies presented in Sect. 4.8: (i) a user’s experience-based (UEB) selection; (ii) a one-at-a-time (OAT) selection; (iii) a selection based on the Morris screening procedure (MR-OAT) and (iv) an approximate POD (APOD) procedure. In the cases (ii)-(iv), the selection criteria were based on the parametric sensitivities of the viscous energy output
J1 computed all over the domain. This choice has been made before selecting more accurate
outputs and observation regions for sake of optimization, to measure the interplay between global shape deformations and the flow field across the domain. Moreover, the choice of 7 control points (denoted with a cross in Fig. 5.25) was excluded due to geometric constraints, since we experienced strong correlations between the displacement of these control points and the variation of the diameters of the inflow and outflow sections, which have to be considered as prescribed in this application. −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Figure 5.25: Reference domain Ω, “truth” finite element mesh, and selected FFD control points used to model the displacements of the shape by means of the (i) UEB, (ii) OAT, (iii) MR-OAT and (iv) APOD procedures (from left to right, from top to bottom). Control points indicated with a cross are not allowed for selection. Vertical displacements πi∈ [−0.2, 0.2] of the points
denoted with a bar are allowed. Stems are referred to parametric sensitivities (see Sect. 4.8) of the viscous energy output J1, scales are not uniform among the procedures (ii)-(iv).
In any case, vertical displacements were found to be uniformly more influent than horizontal displacements. The vertical displacements of the control points selected by means of the procedures
(i)-(iv) are different. By constructing in each of these cases the RB approximation of the flow
problem and solving a shape optimization problem with the viscous energy cost functional J1, we
found a reduction ∆J1 which ranges between 18% (UEB case) and 44% in the APOD case; in
the OAT and MR-OAT cases reduction was rather similar and about 40%, thus leading in this case to a remarkable improvement with respect to the UEB choice.
We thus consider the FFD mapping obtained through the APOD selection procedure (iv) and the computational reference domain represented in Fig. 5.25.
The admissible parameter range for each control point displacement is π ∈ Dπ= [−0.2, 0.2]6 for
i = 1, 2, . . . , 6. A seventh parameter ω ∈ Dω= [0, ωmax] takes into account the magnitude of the
residual flow – and thus the flow splitting. The inflow velocities on Γc (bypass) and Γin (blocked
host artery) are given by a Poiseuille profile vc(ω) and a Gaussian profile vin(ω), respectively.
The dependence of the two flows from the uncertain parameter ω ∈ Dω= [0, ωmax] is such that
the downfield flowrate is constant, with a flow split ranging from 1/0 (complete occlusion of the host artery, for ω = 0) to 2/1 (flowrate through the occluded artery equal to one half of the flowrate through the graft, for ω = ωmax= 15). Moreover, by choosing physiological values for
physical flow viscosity and density we end up with a steady incompressible Navier-Stokes flow characterized by Re = 150 at the location of the anastomosis.
Before constructing the RB approximation of the flow problem, we need to recover the affinity assumption through the EIM procedure. By setting εEIM
tol = 10−4, we obtain an affine expansion
of Qa+ 2Qb+ Qc = 289 terms. The resulting problem is discretized with NV + NQ ≈ 16,000
degrees of freedom, usingP2/P1 finite elements; the dimension of the computed reduced basis
space is 3N = 60 by choosing a tolerance εRB
tol = 5 · 10−2 for the greedy procedure.
In Fig. 5.26 the results of the shape optimization problem for each cost functional Ji, i = 2, 3, 4
are reported, in case of fixed degrees of occlusion; indeed, nopt= 16 shape optimization problems
have been solved, corresponding to the cases ω = 0, 1, . . . , 15. We can notice that the minimum values obtained with the vorticity functional J3and the functional based on Galilean invariant J4
are decreasing function with respect to ω, an indication that the case ω = 0 is the most difficult one concerning vorticity minimization.
0 5 10 15 480 500 520 540 560 580 600 620 640 660 680
Optimal values of functional J1
ω 0 5 10 15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55
Optimal values of functional J2
ω 0 5 10 15 750 800 850 900 950 1000 1050 1100 1150 1200 1250
Optimal values of functional J3
ω
Figure 5.26: Minimum values for the three cost functionals (from left to right: vorticity J3,
Stokes-tracking J2, Galilean invariant J4) as a function of the unknown residual flow parameter
ω. Results obtained by shape optimization for nopt= 16 different cases between ω = 0, 1, . . . , 15.
As expected, the condition leading to the strongest development of vorticity cores is that corresponding to a complete occlusion, for which the flow through the bypass starts creating a strong vortex in the proximity of the anastomosis. As soon as a small residual flow crosses the occluded branch, the vortex cores are removed. On the other hand, also the tracking-type functional detects the presence of the vortex core for small values of ω, but for larger values of ω the misfit between the two flows increases considerably due to the presence of the strong nonlinear convective term. Moreover, the reduction of the cost functional for the three cases ranges from 53% to 73%, averaging the results on the nopt= 16 problems for each case (see Table 5.7).
In Fig. 5.27 the velocity fields for the initial and the optimal shapes obtained with the three cost functionals are reported, considering the minimum (ω = 0) and the maximum (ω = 15) magnitude of the residual flow through the occluded branch. The vorticity cores are clearly observable in presence of a complete artery blockage; moreover, although we get a sensible reduction of the cost functional also in this case, the vorticity cores never disappear completely.
∆J (average) # optim. iters # output evals # total output evals CPU time
J3 0.5297 3 ÷ 7 28 ÷ 99 771 5.9 h
J2 0.6618 3 ÷ 7 28 ÷ 60 641 9.4 h
J4 0.7340 3 ÷ 27 28 ÷ 230 1 120 11.3 h
Table 5.7: Results of the shape optimization problem (nopt= 16 cases, ω = 0, 1, . . . , 15). Here
Figure 5.27: Velocity magnitude [cm/s] and streamlines of the steady incompressible Navier- Stokes (RB simulation); from left to right: initial and optimal configurations obtained with shape optimization of Ji, i = 3, 2, 4, using the two extremal values of the residual flow parameter (ω = 0
on top, ω = 15 on bottom).
Concerning the solution of a robust SO problem under the form (RD-SOµ), we consider only the case of the functionals J3 and J4, since the behavior of the cost functional J2may be influenced
more by the presence of the nonlinear convection than by the vorticity patterns for large values of ω. As in the case of aorto-coronary bypass grafts, we find that the robust configurations correspond, for both cost functionals, to the optimal shapes computed for ω = 0 in the previous case (see Fig. 5.27, columns 2 and 4). As in the case discussed in Sect. 5.4.2, the solution of the robust (shape) optimization problem requires a CPU time which is at least one order of magnitude larger than that used for a (shape) optimization problem (see Table 5.8).
# optim. iters # output evals # total PDE solves CPU time
J3 7 87 1,482 4.0 h
J4 27 230 3,684 19.8 h
Table 5.8: Numerical details for robust shape optimization. Here the number of output evaluations refers to the solution of the minimization problem (for the max Ji(µ) functional).
Here robust shape optimization problems took about 4 ÷ 20 hours of computational time by using the RB method for the fluid simulation. Solving the same problem with the full FEM simulation would have taken considerably more (about 700 hours) if shape deformations had been handled by standard methods like local boundary variations and not taking advantage of the geometrical reduction afforded by the proposed shape parametrization.
We close this section by pointing out that in both cases we have considered, we found that a design that is robust over the entire range ω ∈ [0, ωmax] of (parametrized) residual flows through
the occluded artery must be adjusted mainly for the case of total occlusion, i.e. ω = 0.
In the case that total occlusion of the artery is not expected, we can exclude some region near the point of total occlusion and instead consider the range of uncertainty ω ∈ [ωmin, ωmax] for some
ωmin> 0. In the typical case that the amount of occlusion changes over time, it is imperative
that we should be able to monitor the amount of occlusion over time by simple noninvasive measurements to guarantee that, in fact, ω ≥ ωmin, since otherwise our bypass design will not
be robust any more. This leads us naturally to consider the inverse problem of determining the magnitude of the residual flow, which is discussed in Sect. 6.5 of next chapter.