4.8 Design parameter space reduction
4.8.2 A practical example
We now consider the reduction of parameter space through the selection of a suitable set of control points in a FFD case defined over a starting lattice of 6 × 4 control points. Our goal is the selection of p = 6 design parameters µ1, . . . , µ6for constructing a FFD map, among all the
possible p0 = 2 × 6 × 4 horizontal/vertical displacements of the control points in the lattice (we refer to Fig. 5.25 for the geometrical configuration). The application related to this geometrical configuration deals with the shape optimization of a bypass graft and is discussed in Sect. 5.4.3. The quantity of interest chosen for the parameter screening is the energy functional
S(v) =ν
2 Z
Ω
|∇v|2dΩ,
in order to take into account the interplay between global shape deformations and the flow field across the domain. The results obtained by applying the OAT and the MR-OAT procedures are shown in Fig. 4.3-4.4. To get the reduced set of parameters, 2p0 – respectively R(p0+ 1) – FE approximations have been required by OAT and MR-OAT procedures, being R = 10 in the MR-OAT case. In both cases, horizontal deformations effects were found to be negligible compared to the ones induced by vertical displacements of control points, so that only p0/2 = 24
effects corresponding to vertical displacements of control points are reported.
More effective parameters were found to be rather the same in both cases, even if the MR-OAT procedure provides also information on whether any significant interaction among the parameters exists. Concerning the APOD procedure, largest total correlations arg maxi{∆mini , ∆maxi } and
singular values of the matrix YH are represented in Fig. 4.5 and decay at a similar rate.
0 5 10 15 20 25 0 2 4 6 8 10 µi
Absolute sensitivity of the output function
Figure 4.3: Observed parametric sensitivities (OAT procedure). Here only p0/2 = 24 effects
corresponding to the (more relevant) vertical displacements of control points are represented.
FFD parametrizations originating from these selection procedures have been implemented in Sect. 5.4.3, taking into account some additional constraints related to the application. The subsets of active control points are represented in Fig. 5.25.
0 5 10 15 20 25 0 1 2 3 4 µi
Absolute sensitivity of the output function
Average sensitivity Min sensitivity Max sensitivity
Figure 4.4: Absolute mean effects, minimum/maximum effects (MR-OAT procedure). Here only
p0/2 = 24 effects corresponding to vertical displacements of control points are represented.
0 5 10 15 20 25 10−4 10−3 10−2 10−1 Ordering Singular values K−correlation factors Singular values of YH
Figure 4.5: Convergence of the 24 Q-correlation factors arg maxi{∆mini , ∆
max
i } for a random
sample of 100 snapshots (APOD procedure).
We close this section by pointing out a further issue related with reduction of parameter space dimensions. Dealing with shape parametrizations, often the effect of a single parameter can be quite small (e.g. because of local shape deformation), but their combined effect is large. In this case a reduced ˜µ-parametrization can be obtained from the starting µ0-parametrization, being
µ0∈ D0 ⊂Rp0, by means of a suitable transformation – finally, by a change of variables ˜µ =U(µ0)
in the parameter space, so that ˜µ ∈ ˜D ⊂ Rp˜, where ˜p p0 andU ∈ Rp×p˜ 0 is an orthogonal
matrix.
A method recently proposed by Constantine and Wang [66] for input subspace detection is based on the introduction of the following correlation matrix Cp0×p0, whose components are given by
Cij= Z D0 ∂s(µ0) ∂µ0j ∂s(µ0) ∂µ0i dµ ≈ |D0| R R X r=1 δj(µj,r)δi(µi,r), (4.51)
where the approximation follows by (4.50). The quality of this approximation is crucial in order to represent the variations of the sensitivities over D, and it can be controlled by the number of samples R. Further investigations are ongoing, in order to make this procedure more robust.
0 2 4 6 8 10 10−20 10−15 10−10 10−5 100 105 0 2 4 6 8 10 12 10−20 10−15 10−10 10−5 100 105 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8
Figure 4.6: Top: eigenvalues decay for approximated correlation matrices (4.51) in the OAT and MR-OAT cases. Bottom: components of the first eigenvector in both cases.
By computing the eigenvalue decomposition C = ZΛZT, it is possible to show [66] that the ˜p
(dominant) eigenvectors related to the ˜p largest eigenvalues indicate the dominant directions of
output variation. Thus, a good choice isU = [z1| . . . |zp˜].
Moreover, if the decay of the eigenvalues λ1≥ . . . ≥ λp0 is rapid (typically exponential) a very
low-dimensional subspace may be sufficient to capture most of the variation of the output s(µ0). In Fig. 4.6 we represent the eigenvalues decay obtained by the approximation (4.51) for the OAT and MR-OAT cases (in the former case R = 1 and the absolute mean effects are replaced by the parametric sensitivities (4.49)). In this case retaining ˜p = 1 eigenvectors is enough; moreover,
from Fig. 4.6 it is possible to see that in this (lucky) case the components of the first eigenvector return the most effective parameters already selected (see Fig. 4.3-4.4).
Part III
Applications
5
Optimal design of cardiovascular
prostheses
In this chapter we present the main application driving the development of the reduced framework introduced and analyzed in the previous parts. Our focus is on the reduced solution of optimal design problems related with cardiovascular prostheses, such as bypass anastomoses. We introduce the main features related to the optimal design of cardiovascular prostheses, then we give a brief description of the physiology of the cardiovascular system and the most important arterial diseases such as atherosclerosis, highlighting the strong correlation between blood flows and vessel geometrical configurations. Then, after providing a general framework for modeling the optimal design of end-to-side bypass anastomoses we recall the main results achieved in the last decades and we show how the reduced framework discussed in the previous parts can be applied in order to face this problem. We present some results concerning (i) a simplified version of an optimal flow control problem, (ii) two shape optimization problems dealing with Stokes and Navier-Stokes flows and (iii) a robust shape optimization problem of particular interest in this framework.
5.1
Cardiovascular physiology and arterial pathologies
Cardiovascular diseases are the first cause of death in developed countries and nowadays several methods and tools provided by mathematical modeling and scientific computing prove to be very helpful in improving our understanding of the physiology of the cardiovascular system, as well as of the development of pathological processes. Thanks to the results achieved by computational fluid dynamics in several engineering contexts, numerical simulation of blood flows have become widespread within the bioengineering and medical research community. This is mainly due to (i) the availability of increasing computational power, (ii) the progress in imaging and geometry extraction/reconstruction techniques [213] as well as (iii) the availability of more and more efficient numerical algorithms. For a general introduction to modeling and simulation of the circulatory system we refer to the work by Quarteroni, Tuveri and Veneziani [259] and to some recent books [9, 100] which collect many contributes spanning over a wide range of topics.
The driving factor behind this development is the awareness that numerical models can provide quantitative descriptions of blood behavior in important vascular districts or in vessel networks, and to explain and assess the relationships between vessels shape, haemodynamics, and a family of clinical indicators. Among the latters we mention wall shear stresses, vorticity, viscous energy dissipations: they can be correlated at various extents to the risk of failure in bypass grafting [193], or artery occlusion in presence of stenosis [241], or the one of aneurysm rupture [222]. Such kind of analyses can also help in understanding how different surgical solutions may affect blood circulation.